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FIRST  YEAR  ALGEBRA 


BY 


WEBSTER   WELLS,  S.B. 

AUTHOR    OF    A    SERIES    OF    TEXTS   ON    MATHEMATICS 
AND 

WALTER   W.    HART,  A.B. 

ASSISTANT    PROFESSOR   OF   MATHEMATICS,    UNIVERSITY    OF    WISCONSIN 
COURSE    FOR   THE    TRAINING    OF   TEACHERS 


D.  C.  HEATH   &   COMPANY 

BOSTON       NEW  YORK       CHICAGO 


/^ 


'%^i 


Copyright,  191 2, 
By  D.  C.  Heath  &  Co. 

EDUCAnON  DfclFK, 

IE  2 


.0-. 


PREFACE 

Ix  this  text  the  authors  have  endeavored  to  present  a  course 
in  algebra  for  the  first  year  of  high  school  which  shall  be 
simple,  comprehensible  to  the  students,  and  of  high  educa- 
tional and  mathematical  value. 

They  have  made  the  solution  of  equations  and  problems  the 
core  of  the  course ;  they  have  emphasized  the  essentials, 
avoiding  little-used  complexities  of  algebra;  they  have  taught 
new  ideas  inductively ;  they  have  emphasized  thoughtful  rather 
than  mechanical  solutions  of  exercises  ;  they  have  tried  to 
make  the  course  maintain  and  increase  the  student's  efficiency 
in  arithmetic ;  they  have  tried  to  make  the  course  interesting 
by  including  varied  problem  material  and  historical  notes,  and 
valuable  by  including  practical  applications. 

The  essential  features  of  the  course  have  been  tried  out  in 
the  classroom  by  many  teachers. 

The  text  contains  sufficient  material  to  meet  the  needs  of 
schools  whose  pupils  have  studied  algebra  before  entering  the 
high  school ;  the  topics  have  been  arranged,  however,  so  that 
a  class  may  easily  cover  the  essentials  of  the  course  in  one 
school  year. 

Attention  is  directed  to  the  following  devices  that  have  been 
employed  to  attain  the  desired  ends : 

Each  topic  that  is  taken  up  is  used  in  the  solution  of  equa- 
tions. (See  §§9,  10,  12,  41,  51,  60, 107,  etc.)  This  makes  the 
study  of  the  various  topics  purposeful,  allows  for  good  grada- 
tion in  the  book  as  a  whole,  and  emphasizes  the  equation. 

Problems  are  introduced  at  short  intervals.  Informational, 
geometric,  and  physics  problems  in  reasonable  number  are 
used.  New  types  of  problems  are  introduced  gradually,  ap- 
pearing first  in  classified  lists,  are  taught  with  extreme  care, 
and  are  used  thereafter  in  miscellaneous  lists.  Experimental 
verification  is  suggested  for  some  of  the  facts  from  geometry 
and  physics  that  are  used.  (See  Exercises  7,  25,  28,  29,  38,  39, 
49, 106;  §§  13,  142,  143,  190,  etc.) 

54f;no 


iv  PREFACE 

The  abstract  drill  examples  are  simple,  very  carefully  graded, 
and  numerous.  More  difficult  examples  are  provided  in  an 
appendix. 

Factoring  is  simplified  by  omitting  until  a  later  chapter, 
XVI,  the  more  general  forms  of  the  simple  types  that  alone 
are  needed  in  the  solution  of  equations.  If  desired.  Chapter 
XVI  may  be  studied  immediately  after  Chapter  VIII. 

The  chapter  on  square  root  and  radicals  contains  only  such 
topics  as  are  required  in  the  solution  of  quadratic  equations. 
All  of  the  subject  of  imaginaries  that  is  necessary  is  given  in 
the  chapter  on  quadratics.  Eadicals  and  imaginaries,  as  such, 
are  beyond  the  proper  scope  of  the  first-year  course,  and  will 
be  provided  for  in  a  later  text.  Only  a  brief  chapter  on  ex- 
ponents is  included,  and  that  is  at  the  end  of  the  book. 

To  encourage  thoughtful  solution  of  exercises,  mechanical 
processes  like  "  transposition  "  and  "  clearing  of  fractions  "  are 
not  introduced  until  the  student  is  familiar  with  the  principles 
underlying  these  processes. 

Special  attention  is  given  to  inductive  development  in  Chap- 
ters II,  IV,  and  VIII. 

Checking  is  emphasized  in  a  reasonable  manner  throughout 
the  book. 

To  maintain  and  increase  efficiency  in  arithmetic,  fractional 
and  decimal  coefficients  and  some  large  numbers  are  intro- 
duced —  in  a  sane  wa}^  it  is  believed.  Students  are  encouraged 
to  express  roots  of  equations  in  approximate  decimal  form, 
especially  in  Chapter  XV.  Some  short  cuts  in  arithmetic  are 
included  in  Chapter  VIII. 

The  formula  is  introduced  as  being  essentially  practical. 
(See  §§  17  and  146.)  For  other  applications,  see  §§  44,  84, 
143,  150,  190. 

The  data  for  the  informational  problems  is,  in  the  main,  of 
permanent  rather  than  temporary  interest,  and  of  general 
rather  than  local  interest. 

Graphical  representation  is  introduced  in  a  chapter  preceding 
simultaneous  equations.  The  statistical  data  for  graphical 
representation  contains  only  two  or  three  significant  figures. 


CONTENTS 


I. 
ir. 

111. 


IV 


VI. 


vu. 


Literal  Numbers  .        .        .        . 

Positive  and  Negative  Numbers 

Addition  of  Positive  and  Negative  Numbers 
Multiplication  of  Positive  and  Negative  Numbers 

Addition  and  Subtraction  of  Algebraic  Expres 
SIONS  ........ 

Addition  of  Monomials  .         . 

Addition  of  Polynomials        .... 

Subtraction 

Subtraction  of  Positive  and  Negative  Numbers 
Subtraction  of  Polynomials  .         .        .         , 


Parentheses   . 

Removal  of  Parentheses 
Introduction  of  Parentheses 


Multiplication 

Multiplication  of  Monomials 

Multiplication  of  Polynomials  by  Monomials 

Multiplication  of  a  Polynomial  by  a  Polynomial 


Division 

Division  of  a  Monomial  by  a  Monomial 
Division  of  Polynomials  by  Monomials 
Division  of  Polynomials  by  Polynomials 


Simple  Equations  .... 
Properties  of  Equations 

VIII.    Special  Products  and  Factoring 
Quadratic  Equations  by  Factoring 


VI 


CONTENTS 


PAGE 

IX.     Highest   Common    Factor    and    Lowest   Common 

Multiple 154 

Highest  Common  Factor 154 

Lowest  Common  Multiple 157 

X.     Fractions lilSO 

Reduction  of  Fractions 160 

Addition  and  Subtraction  of  Fractions      .         .         .  169 

Multiplication  of  Fractions. 175 

Division  of  Fractions 178 

Complex  Fractions 182 

XL     Simple  Equations  {Continued) 185 

Fractional  Equations 185 

Solution  of  Literal  Equations 200 

XII.     Graphical  RepiJesentation 206 

XIH.     Simultaneous  Linear  Equations     ....  221 

Literal  Simultaneous  Equations         ....  239 

Equations  containing  Three  Variables      .         .         .  341 

XIV.     Square  Root  and  Quadratic  Surds      .        .        .  244 

Quadratic  Surds 252 

XV.     Quadratic  Equations 254 

Complete  Quadratic  Equations 258 

Quadratic  Equations  containing  Two  Unknowns      .  275 

Imaginary  Roots  in  a  Quadratic  Equation         .         .  278 

XVI.     Special    Products     and    Factoring    (Advanced 

Topics)   .        •. 283 

Factoring  Polynomials 286 

XVII.     Ratio,  Proportion,  and  Variation         .        .        •  290 

Properties  of  a  Proportion           .         .         .         .         .  293 

Variation 299 

XVIII.     General  Powers  and  Roots 304 


FIRST   YEAR  ALGEBRA 


ALGEBRA 


INTRODUCTION 

Algebra  is  like  arithmetic  in  some  respects.  Arithmetic 
consists  of  the  study  of  addition,  subtraction,  multiplication, 
and  division  of  some  kinds  of  numbers,  and  of  the  apx)lication 
of  this  knowledge  to  some  of  the  common  problems  of  daily 
life  and  of  business.  Algebra  continues  this  study  of  numbers. 
In  arithmetic,  numbers  are  represented  by  the  digits  1,  2,  3, 
etc. ;  sometimes  also,  they  are  represented  by  letters,  as,  for 
example,  in  interest  problems,  where  tlie  principal  is  repre- 
sented by  P,  the  rate  per  cent  by  i2,  and  the  interest  by  /. 
These  letters  make  it  possible  to  abbreviate  rules ;  thus,  the 
rule  "  the  interest  for  one  year  equals  the  principal  multiplied 
by  the  rate  per  cent,"  may  be  expressed  by  the  letters  as 
follows : 

100 

In  algebra,  letters  are  regularly  employed  to  represent  num- 
bers. Some  new  kinds  of  numbers  and  many  new  mathemati- 
cal ideas  are  studied,  and,  as  in  arithmetic,  some  of  the  uses 
of  this  knowledge  are  illustrated. 

Algebra  has  a  very  long  history.  A  little  was  known  about 
it  centuries  before  the  Christian  Era.  The  oldest  mathemati- 
cal book  which  we  have,  written  by  an  Egyptian  named  Ahmes, 
contains  some  problems  similar  to  those  found  in  our  algebras. 
Ahmes  lived  before  1700  b.c.  Knowledge  of  algebra  grew 
very  slowly  indeed  for  many  centuries;  in  fact  it  was  not 
until  the  sixteenth  century  that  algebra  assumed  the  form 
which  it  has  to-day,  and  since  then  many  discoveries  and  im- 

1 


2  "        ' '     '  '      '  ■  ALGEBRA 

provements  in  it  have  been  made.  Many  of  the  wisest  mathe- 
maticians of  former  days  contributed  to  this  growth.  Thanks 
to  their  combined  achievements  and  ingenuity,  it  is  now  possi- 
ble for  any  boy  or  girl  in  the  first  year  of  high  school  to  get  a 
much  broader  view  of  the  elementary  part  of  the  subject  than 
many  of  these  men  had. 

Scattered  through  the  text,  will  be  found  historical  notes 
calling  attention  to  some  of  the  epoch-making  innovations  in 
the  development  of  algebra,  together  with  the  name  and  time 
of  the  man  making  the  step  forward. 

I.    LITERAL   NUMBER 

1.  In  arithmetic,  numbers  are  represented  by  the  digits:  1^ 
2,  3,  4,  5,  6,  7,  8,  9,  and  0,  and  combinations  of  them.  In 
Algebra,  numbers  are  also  represented  by  letters.  Numbers 
represented  by  letters  are  called  Literal  Numbers.  The  follow- 
ing examples  illustrate  the  use  of  letters  as  numbers. 

Example  1.  If  a  boy  saves  5  cents  per  day,  how  much  does 
he  save : 

(a)  in  3  days  ?  (5)  in  5  days  ? 

(c)  in  any  number  of  days  ? 

This  last  result  may  be  expressed  by  saying,  "as  many  cents  as  are 
obtained  by  finding  the  product  of  the  number  of  days  and  5." 

In  algebra,  it  may  be  expressed  thus : 

Let  n  =  the  number  of  days. 

Then,  5  x  w  =  the  number  of  cents  saved. 

So,  if  w  is  6,  5  X  n  is  5  X  6  or  30  ; 
.  if  w  is  8,  5  X  71  is  5  X  8  or  40. 

Example  2.     How  many  inches  are  there  : 

(a)  in  9  feet  ?  (?>)  in  any  number  of  feet  ? 

Let  X  =  the  number  of  feet. 

Then,  12  r  =  the  number  of  inches  in  x  feet. 

12  X  is  read  "  twelve  ic." 


LITERAL   NUMBER  3 

2.  Sign  of  Multiplication.  The  symbol,  x,  is  used  to  in- 
dicate multiplication  in  algebra  as  well  as  in  arithmetic ;  it  is 
read  "a'mes"  or  ^^ multiplied  by"  A  dot,  •  ,  placed  above 
the  line,  is  also  used  as  a  sign  of  multiplication,  and  gener- 
ally even  the  dot  is  omitted,  so  that  12  x  m  may  be  written 
12  •  m  or  12  771.  a  x  b  may  be  written  a  •  b  or  abj  and  is 
read  "  a  b." 

Historical  Notk. — The  symbol,  x,  was  first  used  by  an  English- 
man, Oughtred,  about  l()3l.  The  symbol,  •  ,  was  introduced  by  Leibnitz 
in  1693.  Multiplication  was  indicated  as  early  as  the  thirteenth  century, 
in  Hindu  and  Italian  books,  by  simply  writin<j  the  factors  side  by  side. 
This  method  was  forgotten  for  a  time,  and  was  reintroduced  by  German 
algebraists  during  the  fifteenth  century. 

3.  The  result  obtained  by  multiplying  two  or  more  numbers 
together  is  called  the  Product,  and  the  numbers  are  called 
the  Factors  of  the  product. 

EXERCISE  1 

1.  What  does  10  rf  mean  ?     5/-?     6  s? 

2.  How  much  is  10  d,  when  dis2?     3?     5?     6? 

3.  Howmuch  is  7r,  when'ris4?     6?     12?     4? 

Another  way  of  expressing  this  example  is  to  say  :  "  what  is  the  value 
'  of  7  /•  when  r  is  4  '.'  *■ 

4.  Find  the  value  of  8  a  when  a  is  5 ;  15 ;  2.5 ;  |. 

5.  Find  the  value  of  9  TF  when  WisS;  12 ;  ^. 

6.  Tf  a  equals  the  number  of  inches  in  the     a        a        B 


line  AB,  what  does  3rt  e([ual?     Illustrate  it. 

7.  If  b  represents  the  number  of  square  feet  in  a  rectangle, 
what  does  2  b  represent  ?     J  6  ?     6  6? 

8.  If  a  man  earns    S3  per  day,  what  will   he   earn   in  6 
days?  in  20  days?  in  n  days? 


^^r' •''••«''  ':  :..r/.\  ...'        ALGEBRA 

9.   If  a  book  costs  75^,  what  will  three  of  them  cost?     x  of 
them  ?     How  much  is  75  x  j^,  when  ic  is  4  ? 

10.  If  a  train  travels  at  the  rate  of  25  miles  per  hour,  how 
far  will  it  go  in  3  hours  ?  in  5  hours  ?  in  x  hours  ? 

11.  One  cubic  foot  of  water  weighs  62.5  pounds.  How 
much  do  X  cubic  feet  weigh  ?  How  many  pounds  are  62.5  it- 
pounds  when  ic  is  4  ?  5  ?  10  ? 

12.  If  a  farm  consists  of  85  acres,  valued  at  A  dollars  per 
acre,  what  is  the  value  of  the  farm?  What  is  it  when  A 
is  75? 

13.  If  a  man  receives  y  dollars  per  week,  how  much  will  he 
receive  in  a  year?     Find  the  amount  if  y  is  22. 

14.  If  each  of  35  persons  contributes  s  dollars  to  the  expense 
ot  an  excursion,  what  is  the  total  expense?  Find  it  when  s 
is  5 ;  6. 

15.  If  w  represents  a  number,  what  will  represent  a  number 
3  times  as  large?  5  times  ?  2\  times?  Find  the  value 
of  each  of  these  when  n  is  2  ;  6 ;  10. 

4.  The  symbols  (  )  are  called  parentheses.  In  mathematics, 
they  mean  that  the  numbers  within  are  to  be  combined  as  the 
signs  indicate,  and  that  the  result  is  to  be  treated  as  a  whole. 

Thus,  (3  X  6)  —  (7  +  3)  means  :  multiply  3  by  5  ;  add  7  and  3  ;  subtract 
the  second  result  from  the  first. 

5.  An  Important  Multiplication  Law.  When  several  numbers 
are  to  be  multiplied  together,  the  product  may  be  found  by 
multiplying  the  first  factor  by  the  second,  that  result  by  the 
third,  and  so  on. 

Example  1.    2x3x4x5=(2x3)x4x5  =  6x4x5 

=  (6  X  4)  X  5  =  24  X  5  =  120. 

Example  2.  If  a  square  contains  4  a  square  feet,  find  the 
area  of  a  square  5  times  as  large. 


LITERAL   NUMBER  5 

Solution  :  1.  The  area  is  5  x  4  a  square  feet  or  (5  x  4)  x  a  =  20  x  a 
=  20  a  sq.  ft. 

2.  This  result  is  true  for  any  value  of  a  ; 
if  a  =  3,  4  a  =  12, 

and  6  X  4  fl  =  6  X  12  =  60 ; 

also  20  a  =  20  X  3  =  60. 

Tlie  fact  that  tlie  result  is  60  in  both  cases  shows  that  the  solution  is 
probably  coi-rect. 

EXERCISE  2 

Find,  as  in  Examples  1  and  2,  the  results  in  the  following  examples,  and 
test  the  results  as  in  2  for  some  particular  value  of  the  literal  number  : 

1.  6x8a.  3.   9x8><.  5.   3x|m.         7.      8xJ«. 

2.  7x106.         4.    9x172.        6.   5xfA:.  8.    12x|a;. 

9.   If  one  number  is  represented  by  2  6,  what  will  represent 
a  number  3  times  as  large  ?  one  third  as  lai'ge  ? 

10.  If  John  is  4  times  as  old  as  James,  and  if  James  is  2?/ 
years  of  age,  how  old  is  John?    Find  both  ages  if  y  is. 3. 

11.  If  the  volume  of  a  sphere  is  163 1  cubic  inches,  what  is 
the  volume  of  a  sphere  3  times  as  large?     ^ 

12.  If  the  interest  on  a  sum  of  a  money  is  25  r  dollars  for  one 
year,  what  is  the  interest  for  4  yeai'S ?  3  years?  6  years  ? 

13.  There  are  three  numbers  of  which  the  first  is  4  times 
the  second,  and  the  third  is  3  times  the  first.  Kepresent  the 
second  number  by  s,  and  find  the  others.  Find  their  values 
wben  the  number  s  is  5. 

14.  There  are  three  numbers  of  which  the  second  is  8  times 
the  first,  and  the  third  is  4  times  the  second.  Let/  represent 
the  first,  and  then  represent  the  others. 

15.  The  value  of  A's  property  is  5  times  that  of  B's,  and  the 
value  of  C's  property  is  4  times  that  of  A's.  Represent  the 
number  of  dollars  B  possesses  by  6,  and  then  represent 
the  number  of  dollars  owned  by  A  and  C. 


6  ALGEBRA 

6.    An  Important  Division  Law. 

Since  2  xoa  =  6a,  then  6  a  -=-  2  =  3  a. 

Similarly,  40  ic  -^  5  =  8  a;,  since  5  •  8  a.-  =  40  a;. 

Rule. — To  divide  the  product  of  an  arithmetical  number  and  a 
literal  number  by  an  arithmetical  number : 
l.«  Find  the  quotient  of  the  arithmetical  numbers. 
2.  Multiply  the  quotient  of  step  i  by  the  literal  number, 

EXERCISE  3 

1.  Divide  each  of  the  following  numbers  by  5 : 

(a)  25  t.  (p)  30  a;.  (c)  45  rs.  (d)  75  y. 

2.  Divide  each  of  the  following  numbers  by  3 : 

(a)  6r.  (b)  30  c.  (c)  42  d.  (d)  54  e. 

3.  Divide  each  of  the  numbers  in  Example  2  by  2. 

4.  What  part  of  36  ly  is : 

(a)  Siv?  (b)  Aw?  (c)  6w?  (d)  Iw? 

5.  What  part  of  44  a^  is  : 

(a)  lloj?     '    (6)  4a;?  (c)  22a;?         (d)  Ix? 

Find  the  following  quotients  : 

6.  39y-r-3.  '         9.   49  6-5-49.  12.  63  s --63. 

7.  96/H-12.  10.    120^-^120.  13.  72t^72. 

8.  81  a;-- 9.  11.   45  r^ 9.  14.  25  a;-- 25. 

Historical  Note. —The  symbol,  -4-,  was  introduced  by  John  Pell 
who  lived  during  the  seventeenth  century. 

7.  Use  of  Literal  Numbers  in  Solving  Problems.  Literal  num- 
bers aid  in  solving  certain  kinds  of  problems. 

Example.  How  long  will  it  take  a  bricklayer  to  lay  38,500 
bricks  if  he  can  lay  3500  in  one  day  ? 


LITERAL  NUMBER  7 

Arithmetical  Solution 

Since  he  can  lay  3500  bricks  in  one  day,  then  in  the  unknown  number 
of  days  he  can  lay  3500  times  that  number  of  bricks.  Since  this  must  be 
38,500,  according  to  the  statement  of  the  problem,  then  the  number  of 
days  must  be  j^qj^  of  38,500  or  H. 

Algebraic  Solution 

Let  n  =  the  unknown  number  of  days. 

Then,  3500  n  =  the  number  of  bricks  laid  in  these  days, 

and  38,500  =  the  number  of  bricks  to  be  laid. 

So,  3500  n  =  38,500. 

Since  one  n  is  ^^^^^  ^^  ^^^  "'  divide  these  two  equal  numbers  by  3500. 
Then,  w  =  11. 

Test:  11  is  correct,  for  3500  x  11  =  38,500. 

8.  The  mathematical  statement  3500  n  =  38,500  is  called 
an  Equation.  The  literal  number  in  the  equation  is  called  the 
Unknown  Number. 

An  Equation  expresses  the  equality  of  two  numbers. 

The  numbers  on  the  right  of  the  equality  sign  form  the 
Right  Member  of  the  equation,  and  the  ones  on  the  left,  the 
Left  Member. 

An  equation  implies  a  question  :  "  for  what  value  of  the  un- 
known number  is  the  equality  true  ?  " 

For  example,  in  the  equation  of  §  7,  n  can  have  only  one  value,  —  the 
one  found,  11 ;  thus,  n  cannot  be  10,  for  3500  x  10  is  35,000,  and  not 
38,500. 

Finding  the  value  of  the  unknown  is  called  Solving  the 
Equation. 

Historical  Note.  — The  equation  is  implied  in  Ahmes'  book.  To  in- 
dicate the  unknown  number,  he  used  a  word  hau  corresponding  to  our 
word  heap,  Diophaiitus,  a  Greek  mathematician  of  the  fourth  century, 
used  for  the  unknown  the  last  letter,  s,  of  the  word  for  number  ;  Vieta, 
a  French  mathematician  of  the  sixteenth  century,  used  the  vowels,  A,  E^ 
/,  O,  [/"and  Y;   Harriot,  an  English  mathematician  of  about  the  same 


S  ALGEBRA 

time,  also  used  the  vowels  but  wrote  them  with  small  letters  ;  Descartes,  a 
French  mathematician  of  the  same  period,  used  the  last  letters  of  the 
alphabet,  x,  ?/,  and  z. 

9.  In  solving  the  equation  in  §  7,  two  equal  numbers  were 
divided  by  the  same  number.  It  is  clear  that  if  equal  numbers 
are  divided  by  equal  numbers,  the  quotients  are  equal. 

This  fact  is  used  in  algebra  in  the  following  form : 

Rule.  —  Both  itiembers  of  an  equation  may  be  divided  by  the  same 
number  without  destroying  the  equality. 

Example.     Solve  the  equation  : 

36  k  =  468. 
Solution:  1.    Since  k  is  ^^  of  36  k,  divide  both  members  of  the  equa- 
tion by  36. 

2.  k  =  -S%^  (Rule  §  9) 

8.  =  13. 

EXERCISE  4 

Solve  the  following  equations  : 

1.  7p=238.  6.  27?/  =  351. 

2.  8n  =  608.                                7.  2  a;  =  161. 
8.    9(c  =  423.                               8.  5v  =  21S. 

4.  6A=  312.  9.  8  m  =  1864. 

5.  16a;  =  240.  10.  10  ?t' =  2345. 

t'he  arithmetical  solution  of  the  following  examples  is  easy.  Their 
algebraic  solution  leads  to  the  simplest  form  of  equation.  Give  the  alge- 
braic solution. 

11.   What  number  multiplied  by  13  equals  221  ? 

i^.  Th^  product  of  a  certain  number  and  17  equals  403; 
IftM  ^he  hUmber. 

IS.  A  farm  consisting  of  43  acres  is  offered  for  sale  at  the 
^rice  $  3655.     What  is  the  average  price  per  acre  ? 

14.   tVhat  number  multiplied  by  3.7  equals  8.51  ? 


LITERAL   NUMBt:R  % 

(15  J  If  the  total  expense  for  a  picnic  for  a  party  of  18  boys 
anoTgirls  is  $ 6.94,  how  much  must  each  one  oontribute  ? 

16,  A  man  is  compelled  to  make  a  journey  of  126  miles  in 
his  automobile  over  a  poor  road  in  7  hours ;  how  many  miks 
mu§t  he  average  per  hour  ? 

UJ  The  fastest  train  on  the  Pennsylvania  Railroad  between 
»^t.  Louis  and  New  York  makes  the  trip  in  24  hours ;  if  the 
distance  is  1052.4  miles,  what  is  the  average  rate  per  hour? 

10.   A  second  rule  used  in  solving  equations  is : 

Rule.  —  Both  members  of  an  equation  may  be  multiplied  by  the 
same  number  without  destroying  the  equality. 

This  fact  may  "be  illustrated  by  the  scales.     Sup- 


^ 


pose  that  the  sugar  S  balances  the  weight  11^;  if 

tlie  weight  is  doubled,  then  the  weight  of   sugar    j*       "     _^„_  J*"""! 

must  also  be  doubled  in  order  to  keep  the  balance. 

Example  1.  The  circumference  of  one  of  the  large  redwood 
trees  of  California  is  70  feet.  Find  its  diameter.  (The  circum- 
ference of  a  circle  is  twenty-two  sevenths  of  its  diajueter.) 

Solution  :   1.     Let         d  =  the  number  of  feet  in  the  diameter. 

99  * 

2.   Then,  ^tZ  =  70,  the  number  of  feet  in  the  circutnferenOfe. 

7 

8.   Multiply  both  members  of  the  equation  by  7. 

Then,  J.^d  =  7.70,  (§10) 

or  22  d  =  490. 

4.   Divide  both  members  of  the  equation  by  22. 

Then,  d  =  ^  =  22t3^,  or  22.2+  feet. 

Check  :  Does         y  of  22^^  =  70  ? 

2        35 
|x22A^f'x*^=70. 


10  ALGEBRA 

Example  2.     Solve  the  equation  ^x=  142. 


2.  Multiply  both  members  of  the  equation  by  8. 

1    . 
Then,  ^  .  5  a-  ^  g  •  142,  (Rule,  §  10) 

or  5  a;  =  1136. 

3.  Divide  both  members  of  the  equation  of  step  2  by  5. 

Then,       ^  x  =  ^W  =  227.2.  (Rule,  §  9) 

K       28.4 
Check  :  Does  -  x  ^*i^=  6  x  28.4  =  142.0  ?    Yes. 


EXERCISE  5 

Solve  the  following  equations  and  problems : 

1.  8a=280.  6.   fa;  =  81. 

2.  15^  =  345.  7.   f2/  =  188. 

3.  27  c  =  1242.  8.    f2!  =  96. 

4.  76  m  =  1444.  G)-^^-t  =  A29. 

5.  27.5  X  =  277.75.  (w)  |  r  =  200. 

11.  Three  tenths   of   tl^e   cost  of  a  certain  automobile   is 
$210.     Find  the  cost  of  the  automobile. 

12.  The   selling  price   of   a  certain   book  is  f  of   its  cost. 
Find  its  cost  if  it  sells  for  $1.50. 

13.  Five   eighths   of   a  certain   number   is   95.     Find    the 
number. 

14.  Thirteen  ninths  of  a  certain  number  is  143.     Find  the 
number. 

15.  Two  fifths  of  the  area  of  Lake  Michigan  is  9200  square 
miles.     Find  the  area  of  Lake  Michigan. 

16.  Three   eighths   of   the    cost    of  the    Suez    Canal   was 
$37,500,000.     Find  the  cost  of  the  canal. 


LITERAL  NUMBER  11 

17.  Seven  twenty-fifths  of  the  distance  from  New  York  to 
San  Francisco  is  910  miles.  Find  the  distance  from  New 
York  to  San  Francisco. 

18.  Many  metal  articles,  like  a  brass  candlestick,  are  made 
by  pouring  melted  metal  into  a  mold.  The  piece  taken  from 
the  mold  is  called  a  casting. 

In  making  a  brass  casting,  .^1^  of  the  metal  is  lost  in  the 
melting.  How  much  brass  must  be  melted  to  make  a  casting 
which  will  weigh  72  pounds  ?     (Find  the  second  decimal.) 

19.  Cottonseed  meal  is  used  as  a  fertilizer  on  farms.  It 
contains  about  7%  of  nitrogen,  a  necessary  plant  food.  How 
many  pounds  Of  cottonseed  meal  must  a  farmer  purchase  who 
wishes  to  distribute  15  pounds  of  nitrogen  over  an  acre  of 
ground  ? 

20.  Tobacco  stems  also  are  used  as  a  fertilizer.  They  con- 
tain about  8  %  of  potash,  another  necessary  plant  food.  How 
many  pounds  of  tobacco  stems  must  a  farmer  purchase  who 
wishes  to  obtain  12  pounds  of  potash? 

11.  Addition  and  Subtraction  of  Numbers  having  a  Common 
Factor. 

A  number  which  is  a  factor   of  two  or  more   numbers  is 
called  a  Common  Factor  of  these  numbers. 
Thus,  3  is  a  common  factor  of  6  and  9. 

rt  is  a  common  factor  of  4  a  and  7  a. 

6  is  a  common  factor  of  3  x  5  and  2  x  6. 

A  short  method  of  adding  numbers  which  have  a  common 
factor  is  illustrated  in  the  following  examples. 

Historical  Note.  — The  symbol,  +,  was  first  used  in  print  by  a  Ger- 
man mathematician,  Widmann,  in  1480.  The  origin  of  the  symbol  is  much 
in  doubt.  Italian  writers  of  this  period  used  the  symbol  p,  the  first  letter 
of  the  Latin  word.  plus.  One  explanation  given  for  the  sign,  4-,  is  that 
it  comes  from  an  inverted  i,  f.  The  Latin  word,  e<,  means  and^  and  in 
place  of  it  this  inverted  t  was  often  used.  It  is  easy  to  see  how  the 
symbol  4-  may  have  been  derived  from  the  symbol,  ?. 


ISi  ALCxEBRA 

EXERCISE  6 

1.   3  times  7  plus  2  times  7  is  5  times  7. 

3x7        +  2x7      -      5x7    ■ 

for  21  +  14        =  35. 

2.  (12  X  9)  +  (8  X  9)=20  x9=  ? 

3.  (8  X  4)  +  (7  x4)  =  (?)x  4  =  ? 

4.  (5x  7)  +  (6x7)  +  (9  x  7)=(?)x7=  ? 

5.  6  times  ?^  -f  4  times  n  =  (?)  times  n  ? 

6.  6.a;-h4.a;=(?)a;?  9.    12  i; -f- 6  v  +  3 -v  =  ? 

7.  7a-f3a=^?  10.   2r4-3r-f-5r  +  10r=  ? 

8.  11 2/4-82/ -f52/  =  ? 

11.    (5  X  4)-(2  X  4)=3  X  4  =  12,  for  20  -  8  =  12. 
13,.  (10  x.7)-(4  X  7)  =  (?)x  7  =  ? 
13.    (12.x  8)-(5  X  8)  =  (?)  x  8  =  ? 

14.  9a>-^5x=:^{?)x?  18.    13  t  -  5  t  +  9  t  -  4:t=r 

15.  16&-55  =  ?  19.   12A  +  eA-2A-^5A=? 

16.  20  2/ -1^2/  =  ^  20.    8r  +  13r-llr-f  5r=? 

17.  4  m  +  6  m  —  2  m  =  ? 

21^    One   number   is   four   times   another.      Eepresent   the 
smaller  by  s.     Then  represent  the  larger  and  find  their  sum. 

22.  One  number  is  |  as  great  as  *an other.     Let  b  equal  the? 
larger.     Eepresent  the  smaller  and  find  their  sum. 

23.  One  number  is  5  times  as  large  as  another.     Let  s  equal 
tHe  smaller*    Represent  the  larger  and  then  find  their  difference. 

24.  The  base  of  a  rectangle  is  three  times 
the  altitude.  Eepresent  the  altitude  by  a ;  then 
rcjpresewt  the  base.     Find  also  the  perimeter. 

(ThSi perimeter  is  the  sum  of  the  lengths  of  the  sides.) 
JJ5.   What  is  the  perimeter  of  a  triangle  if  one  of  its  sides 
is  2  ^  inches,  if  the  second  side  is  three  times  as  long  as  the 
first  and  the  third  side  is  2i  times  as  long  as  the  first  ? 


LITERAL  NUMBER  1$ 

26.  How  many  incbes  in  n  feet  ?  in  n  yaids  ?  in  n  yards  -f 
/^  feet  -I-  n  inches  ? 

27.  How  many  cents  in  x  nickels  ?   in  ic  dimes  ?    in  a?  dimes 
+  X  nickels  -f  a;  cents  ? 

Simplify  the  following : 

28.  2c+ic.       30.    5a;-f-fa'.      32.    2h  +  /ABb.       34.    ^a^  +  J^'. 

29.  .Sc-h|c.       31.    3cH-.2c.     33.    4mf.l5m.      36.    §^-f-|y. 

12.  A4ditioo  of  Literal  Numbers  used  in  ^.quations. 

Example  1.     The  sum  of  two  niimbeiv  's  yi.     The  greater 
number  is  12  times  the  smaller.     Fiud  the  u.>:nbers. 
Solution  :  1.    I^t  s  =  the  smaller  number. 

2.  Then,  12  s  =  the  laj-ger  number. 

3.  Then,  s  +  12  s  =  01 ,  since  tlie  gum  of  the  joumhexs  is  91. 

4.  Adding,  13  s  =  91. 

5.  Dividing,  s  =  7. 

Chkck  :   If  the  smaller  number  is  7,  the  larger  must  be  84  and  thair 
sum  is  Ul . 

EXERCISE  7 
Solve  and  check  the  following  equations : 

1.  3a  +  4a=:42.  7.   3 a; +  11  a; 4- 12 a;  =  130. 

2.  4m  +  5m  =  108.  8.   15^  +  8^-3^  =  20. 

3.  76-6  =  66.  9.    7y-5r  +  6r=*4. 

4.  3x'  +  7a;=120.  10.    18z/;  -  7w  + 9  w  =  65. 
6.    lly-2y  =  Sl.                       11.    22a;  +  13a;  -  6a;  =  116. 
6.    6z  +  52  =  99.                          12.    16t/-3y  +  4y  =  102. 

13.  The  greater  of  two  numbers  is  four  times  the  smaller. 
The  sum  of  the  numbers  is  60..    Find  the  numbers. 

14.  If  five  times  a  certain  number  is  increased   by  three 
times  the  same  number,  the  result  is  168.     Find  the  number. 

15    Divide  $56  between  A  and  B  so 'that.  A  shall  receive 
seven  times  as. much  as  B. 


14 


ALGEBRA 


16.  A,  B,  and  C  together  have  $  96.  B  has  twice  as  much  as 
C,  and  A  has  as  much  as  B  and  C  together.   How  much  has  each? 

17.  A  man  had  $4195.  After  spending  a  certain  sum,  he 
found  that  he  had  left  four  times  as  much  as  he  had  spent. 
How  much  did  he  spend  ? 

18.  The  sum  of  three  numbers  is  120.  The  second  is  five  times 
the  first,  and  the  third  is  nine  times  the  first.    Find  the  numbers. 

^\9J  The  sum  of  thrr  e  numbers  is  360.  The  second  is  four- 
teen times  the  first,  a  id  the  .third  is  the  sum  of  the  other  two. 
Find  the  numbers. 

20.  Three  m*".j:  are  asked  to  contribute  to  a  fund.  The  first 
agrees  to  give  twice  as  much  as  the  second,  and  the  third  to 
give  twice  as  much  as  the  first.  How  much  must  each  con- 
tribute to  make  a  total  of  $  525  ? 

21.  The  perimeter  of  the  triangle  ABC 
is^40  inches.  Find  the  lengths  of  its 
sides. 

22.  The  perimeter  of  a  rectangle  is  132 
inches.  The  base  is  double  the  altitude. 
Find  the  dimensions. of  the  rectangle. 

23.  The  length  of  the  fence  about  a 
rectangular  field  is  320  rods.     If  the  long  dimension  is  three 
times  the  short  dimension,  find  the  length  of  each. 

24.  The  perimeter  of  the  quadrilateral  ABCD  is  220  inches. 
The  side  CD  is  twice  as  long  as  the  side 
AB,  the  side  AD  is  three  times  as  long; 
the  side  BC  equals  the  sum  of  the  sides 
AD  and  CD.     Find  the  length  of  each   b" 
side. 

25.  The  shortest  distance  by  railroad  from  New  York  to 
Chicago  is  10  times  the  distance  from.  New  York  to  Philadel- 
phia. The  sum  of  the  two  distances  is  990  miles.  Find  the 
distance  from  New  York  to  Chicago  and  to  Philadelphia. 


2a 


LITERAL  NUMBER 


15 


PROBLEMS  ABOUT  ANGLES 

13.    When  two  lines  meet  they  form  an  Angle  (Z). 

The  angle  ABC  is  a  Right  Angle. 

Angles  are  measured  by  a  unit  called  a  Degree  (°).     c 

A  right  angle  contains  90°. 

Two  angles  whose  sum  is  a  right  angle  are  Comple- 
mentary Angles ;  each  of  the  angles  is  called  the     c 
Complement  of  the  other.     The  angles  AOB  and 
BOC  are  complementary  ;  hence  a  -f-  6  =  90. 


EXERCISE  8 

1.  How  many  degrees  are  there  in  one  half  of  a  right 
angle  ?     one  third  ? 

2.  What  is  the  complement  of  30°  ?    40°  ?    70°  ?    a°?   x°? 

3.  Are  angles  of  25°  and  55°  complementary  ?     Why  ? 

4.  If  the  angles  3x  and  7x  are  complementary,  what  is 
their  sum  ?  Fqrm  an  equation  and  determine  x.  What  are 
the  angles  ? 

5.  Determine  the  angles  5  a  and  4  a  if  they  are  comple- 
mentary. 

6.  What  angle  is  double  its  complement?  (Let  c  equal 
the  number  of  degrees  in  the  complement;  form  an  equation.) 

7.  What  angle  is  three  times  its  com- 
plement ? 

8.  What  angle  is  five  times  its  com- 
plement ? 

A    Straight  Angle   equals   two    right    a 
angles.  • 

What  kind  of   angles  are  the  angles  AOC    '^  ^  ^ 

and  BOC?     How  many  degrees  in  their  sum  ? 
How  many  degrees  in  Z  AOB  ?     How  many  degrees  in  a  straight  angle  ? 


z:^ 


16  ALGEBRA 

9.   Find  each  angle  in  the  adjoining  figure, 

10.  There  are  three  angles  whose  sum  is  180°. 
The  second  is  double  the  first,  and  the  third  is  the  sura  of  the 
other  two.  Draw  a  figure  to  illustrate  this  problem.  Find 
the  angles. 

Two  angles  whose  sum  is  a  straight  angle  are  called  Supple- 
mentary Angles ;  each  of  th^i  is  called  the  Supplement  of  the 

other. 

11.  What  is  the  supplement  of  50°  ?  90°?   100°?   x°?  2a°? 

12.  The  angles  5  x  and  7  x  are  supplementary      How  many 
degrees  are  there  in  each? 

13.  Find  the  angle  which  is  four  times  its  supplement. 

14.  Find  the  angle  which  is  five  times  its  supplement. 

=  \^ 

$ B  ^^^ 


^ 


Fig.  1  Fig.  2  Fig.  3 

The  sum  of  all  the  angles  around  a  point  is  4  right  angles  or  360^. 

Thus,  a-f?>  +  c4-d-}-e=5  360. 

i.    Find  each  of  the  angles  in  Figure  3. 

16/  There  are  four  angles  whose  sum  is  the  total  angle 
around  a  point.  The  first  angle  contains  a°]  each  of  the 
others  is  double  the  preceding.     Draw  a  figure  to  illustrate  this 

a  em.     How  many  degrees  are  there  in  each  angle  ? 
There  are  four  angles  whose  sun^is  the  total  angle  around 
a  point.     The  second  angle  is  one  half  of  the  first;  the   third 
angle  is  three  halves  of  the  first ;  and  the  fourth  angle  is  four 
times  the  third.     Find  the  angles. 


LITERAL  NUMBER  17 

If  the  angles  of  triangle  ABC  are  torn  off  and  placed  side 
by  side,  their  sum  is  found  to  be  180°. 

Draw  a  triangle  with  a  ruler,  and  me  if 
you  find  this  to  be  true. 

From  this  we   conclude   that   the 
sum  of  the  angles  of  a  triangle  is  180°. 

\JJB/''  The  second  angle  of  a  triangle 
is   double   the   first,    and   the   third 

angle   is   six    times  the  first.     How     /  a^b/cS 

many  degrees  are  there  in  each  ? 

19/  Find  the  angles  of  a  triangle  when  two  of  th^e  angles  ai*e 
equal  and  the  third  is  equal  to  the  sum  of  the  other  two. 

20.    Find  the  angles  of  a  ti-iangle  if  the  first  is  4  times  the 
second,  and  the  third  is  7  times  the  second. 


DEFlNITIONa 

14.  An  Algebraic  Expression,  or  simply  an  Expression,  is  a 
number  expressed  in  algebraic  symbols;  as, 

2,  ab,  2x-3zy,  -- 
s 

The  Numerical  Value  of  an  expression  is  found  by  substitut- 
ing particular  values  for  the  literal  numbers,  and  performing 
'the  indicated  operations. 

ab  indicates  that  a  is  to  be  multiplied  by  b. 

-  indicates  that  r  is  to  be  divided  by  s. 
s 

2x  —  Syz  indicates  that  3  times  the  product  of  y  and  z  is  to  be  sub- 
tracted from  2  times  x. 

15.  If  the  same  number  is  used  as  a  factor  one  or  more  times 
to  form  a  product,  the  resnlt  is  called  a  Power  of  the  number. 

The  number  itself  is  called  the  BAse. 

An  integer  written  at  the  right  of  and  above  the  base,  to 


18  ALGEBRA 

indicate  the  number  of  times  the  base  is  used  as  a  factor,  is 
called  an  Exponent.     Thus, 

a^,  read  "  a  square  "  or  "  a  second  power,''  means  a  x  a  ; 

a^,  read  "  a  cube  "  or  "a  third  power,""  means  a  x  a  x  a  ; 

a*,  read  ^^  a  fourth'''  or  "  a  fourth  power"  means  a  x  a  x  a  x  a. 

If  no  exponent  is  written,  the  exponent  1  is  understood. 

Historical  Note.  —  Mathematicians  sought  suitable  symbols  for  "the 
powers  of  a  number  for  a  long  time.  At  first  words  were  used  for  them. 
Our  "a  square  "  and  "a  cube"  owe  their  introduction  to  Greek  mathe- 
maticians who  called  the  second  power  by  a  word  which  means  the  square, 
and  the  third  power  by  one  which  means  the  cube.  Herigone,  a  French 
mathematician  of  the  early  part  of  the  seventeenth  century,  wrote  a2,aS, 
«4,  etc.,  and  finally  Descartes,  in  1637,  introduced  the  present  symbols. 
The  word,  power,  comes  from  a  Latin  word,  potentia,  which  corresponds 
to  the  Greek  word  used  for  the  second  power.  The  word,  exponent,  was 
introduced  by  Stifel  about  1553. 

16.  The  Fundamental  Operations  are  addition,  subtraction, 
multiplication,  and  division.  Indicated  operations  are  to  be 
performed  in  the  following  order :  first,  all  multiplications  and 
divisions  in  their  order  from  left  to  right;  then  all  additions 
and  subtractions  from  left  to  right. 

Example.     Find  the  numerical  value  of  the  expression, 

4ab  +  —-d% 
b 

when  a  =  4,   6  =  3,   c  =  5,   d  =  2. 

Solution  r   Substituting, 
4a6+  —  -d3  =  4.4.3  +  — -28  =  4.4-3  +  —  -2.2-2 
=  48  +  10  -  8  =  50. 

EXERCISE  9 
Find  the  numerical  value  of  the  following  expressions  when 
a  — 2,    b  =  5,   c  =  3,    cZ  =  4,    m  =  4,   n  =  3. 

1.  3a  +  56.  3.    5m4-3n.  5.    b^  -  d\ 

2.  4c -2d.  4.    a'^-^c^  6.    a^  +  n^ 


7. 

2c' -3b. 

8. 

2d_^3c^ 
m        n 

9. 

3ab-\-2c-id, 

10. 

5b-{-6m-n\ 

11. 

b     d     m 

LITERAL  NUMBER  19 

12.   2a2_3a4-l  . 
X^  m'  —  mn  -\-  n\ 
14)  a3  +  3a26  +  3a62. 
(Tfi)  2aH-4c-3d 

16.   c«H-6«. 

Write  in  symbols  the  following  and  find  their  value: 

17.  The  sum  of  a  and  b  ;  c  and  d ;  m  and  n. 

18.  The  difference  between  a  and  6 ;  c  and  d. 

(The  difference  between  a  and  6  is  6  —  a.) 

19.  The  product  of  a  and  b ;  c  and  d ;  m  and  w. 

20.  The  quotient  of  a  and  b ;  c  and  d ;  m  and  n. 
gl^a  increased  by  2  6 ;  c  increased  by  3  d. 

(2^  The  square  of  m  increased,  by  the  square  of  n, 
(23^  The  cube  of  b  decreased  by  the  cube  of  m. 

24.    10  more  than  3  a ;  5  less  than  4  6. 

25    3  more  than  the  quotient  of  m  divided  by  d 

26.  4  less  than  the  product  of  a  and  c. 

27.  The  sum  of  the  squares  of  a  and  b. 

FORMULAE 

17.    When  a  rule  of  computation  is  expressed  by  means  of 
algebraic  symbols,  the  result  is  called  a  Formula. 

Example  1.     Find  the  formula  for  the  area  of  a  rectangle. 
Solution  :  The  area  equals  the  product  of  the  base  and  altitude. 
Let  a  =  the  number  of  units  in  the  altitude. 
Let  b  =  the  number  of  units  in  the  base. 
Let^  =  the  number  of  square  units  in  the  area; 
then,  A  =  ab. 


20  ALGEBRA 

The  following  examples  show  how  to  use  a  formula. 

Example  2.     Find  the  -area  of  a  rectangle  whose  altitude  is 
8  inches  and  wkose  base  is  15  inches. 

Solution  :  1.     Use  the  formula  A  =  ab. 
2.  Substitute  8  for  a,  and  15  for  6. 

Then,  ^  =  8  x  15  or  120. 

Example  3.     Find  the  altitude  of  a  rectangle  whose  base  is 
75  'feet  and  whose  area  is  675  square  feet. 

Solution  :  1.     Substitute  in  the  formula  A  =  ab. 
A~m^;  b  ~  75. 

2.  Then,  675  =  a  x  75, 
or                            675  =  75  a. 

3.  Divide  both  members  by  75 :  "9  =  a. 

Example  4.     Find  the  base  of  a  rectangle  whose  altitude 
is  11  inches  and  whose  area  is  385  square  inches. 

Solution  :  1.     Substitute  in  the  formula  A  —  ab. 

2.  ^  =385;   a  =  11. 
Then,                      385  =  11  x  6. 

3.  Divide  both  members  by  11  :  W^  =  b, 
or  6  =  35. 

Rule.  —  To  solve  a  problem  by  a  formula : 

1.  For  the  known  letters  in  the  formula  substitute  their  values. 

2.  Perform  all  of  the  indicated  operations. 

3.  If  an  equation  is  formed,  solve  for  the  unknown  letter,  if  pos- 
sible. 

EXERCISE  10 

1.   The  figure  XFZTTis  a  Parallelogram  (O). 

Find  a  formula  for  determining  its  area. 

(a)  Let  b  and  a  equal  the  units  in  the  base  and 
the  altitude,  and  A  the  square  units  in  the  area.  /~~| 7 

(6)  How  does   XYZW  compare  in   area  with      yZ_I_!_fL4 
FGHK?  ^ 

(c)  What  is  FGHK?     What  is  its  base?   ahi-       p 
tude  ?  area  ?  « 


W 


(d)  What  then  is  the  area  of  XYZW?  G 


^ 


LITERAL   NUMBER  21 

(e)  Make  a  rule  for  finding  the  area  of  a  parallelogram. 
(/)  Expressed  as  a  fortuuia,  thie  rule  is 

A=ab 
(g)  Find  A  when  a  =  12  and  b  =  20. 
(A)  Find  A  when  a  =  15  and  6  =  26. 
(0    Find  a  when  ^1  =  500  and  b  =40. 
( j)  Find  a  when  ^  =  600  and  6  =  26.  I 
(A;)  Find  b  when  ^  =  750  and  a  =  16. 
^JFind  b  when  ^  =  9G0  and  a  =  32.      *'' 

2.    The  figure  XFZ  is  a  Triangle  (A  XFZ). 

Find  a  formula  for  determiiiiiig  its  area. 

(a)  Let  6,  a,  and  A  represent  the  units  in  the 
base,  altitude,  and  area  respectively.  >< 

(6)   What   is  the    figure    XiZlV?    what  is   its         /la'^^,.^ 
base  ?  altitude  ?  area  ?  '•^  b  ^ 

(c)  What  part  of  O  JT  TZ  W  is  A  A'  1'^  ?                      /T!\l^  "7^ 
^  (d)  What  then  is  the  area  of  A  XYZ  ?  Y^-i ^^^ 

(e)  Make  a  rule  for  finding  the  area  of  a  triangle. 

(/)  Expressed  as  a  formula,  this  rule  is, 

A  =^ab. 
2 

(g)  Find  A  when  «  =  10  and  b  =  17. 

§Find  A  when  <t  =  20  and  6  =  30. 
!  Find  6  when  A  =  200  and  a  =  40. 
^  Find  a  when  ^  =  1200  and  b  =  60. 

3.   The  figure  of  the  rectangulai*  solid  has  the  dimensions 
indicated. 

Find  a  formula  for  determining  its  volume. 

Let  T"  represent  the  numl)cr  of  units  in  Uic  voittme. 

The  volume  equals  the  product  of  the  three  dimen- 
sions. 

(a)  Express  this  <Hile  as  «  formula. 

(6)  Find  V  when  a  =  3,  6  =  5,  c  =  8. 

J^  Find  V  when  a  =  C,  ?>  =  9,  c  =  7. 
uSl)  Find  c  when  F=  240,  a=(i,  and  6  =  5. 


22 


ALGEBRA 


4.  The  figure  XYZW  is  a  Pyramid. 
Its  volume  equals  one  third  of  the  product 

of  its  base  and  altitude. 

(a)  Express  this  rule  as  a  formula,  letting  F equal 
the  number  of  units  in  the  volume. 
(6)  Find  Fwhen  a  =  1%,  h  =  15. 
(c)  Find  h  when  V=  160,  a  =  24. 
{d)  Find  a  when  V~  900,  h  =  30. 

5.  The  formula  for  the  circumference  of  a  circle  is 

C=2  7ri?, 

where  (7=  the  number  of  units  in  the  circum- 
ference, 

where  i?  =  the  number  of  units  in  the  radius, 
where  7r  =  3.1416  (tt  is  read  "pi"), 
(a)  Express  this  rule  in  words. 
(§^Find,  by  the  formula,  C  when  B  is  10  inches, 
r^c)  ,find,  by  the  formula,  B  when  C  is  628.32  inches 

6.  The  formula  for  the  area  of  a  circle  is : 

A  =^  Trie. 

(a)  Express  this  rule  in  words. 

(6)  Find,  by  the  formula,  A  when  B  is  10  inches. 

fc)  Find,  by  the  formula,  A  when  ^  is  5  feet. 


(  .7.   The  numbers  s,  v,  t,  and  g  are  connected  by  the  formula: 


find  s  when  v 


50,  t 


s=vt-\-^^gf; 
3,  ^  =  32.16. 


wv 


8.  From  the  formula  £*  =  —-,  find  J5;  when  w=:75,v  =  50, 

g  =  32.16.     (Carry  the  result  to  one  decimal  place.) 

9.  From  the  formula  F=|  ttI^^  find  Fwhen  E  is  3. 
10.   From  the  forniula  aS  =  4  ttM^,  find  S  when  E  =  5. 


II.   POSITIVE   AND   NEGATIVE   NUMBERS 

18.  The  first  numbers  studied  in  arithmetic  are  the  integers, 
such  as,  1,  4,  15,  etc.;  the  next  are  the  common  fractions,  such 
as,  ^,  J,  ]f,  I,  etc.  and  the  decimals,  such  as,  2.03,  4.6,  etc.  The 
literal  numbers  in  the  first  chapter  represented  only  these 
same  arithmetical  numbers.  One  of  the  most  distinctive 
things  about  algebra  is  its  use  of  certain  other  numbers. 

19.  Opposite  Quantities.  Suppose  that  the  temperature  at  a 
certain  hour  of  the  day  was  73°,  and  that  there  was  a  change 
of  5°.  To  determine  the  new  temperature,  it  would  be  neces- 
sary to  know  whether  the  change  was  a  rise  of  5  °  or  the  oppo- 
site, a  fall  of  5  °. 

Suppose  that  a  person  was  known  to  weigh  85  pounds,  and 
that  during  a  certain  time  there  was  a  change  in  his  weight  of 
5  pounds.  To  determine  his  new  weight,  it  is  necessary  to 
know  whether  the  change  was  an  increase  of  5  pounds  or  the 
opposite,  a  decrease  of  5  pounds. 

These  are  two  illustrations  of  opposite  quantities.  Many 
concrete  quantities  exist  in  two  such  opposite  states. 

EXERCISE  11 

Tell  the  opposite  of  each  of  the  following : 

1.  Sailing  35  miles  north.       5.    Moving  5  steps  forward. 

2.  Sailing  25  miles  east.         6.    Depositing  $15  in  a  bank. 

3.  Receiving  $30.  7.   Rise  of  10°  in  temperature. 

4.  Gaining  $5.  8    Walking  5  rods  to  the  right. 

9.    Increasing  weight  by  5  pounds. 

10.   Adding  7. 

93 


24  ALGKBRA 

11.  What  is  the  total  result  if  any  one  of  the  changes  in 
Examples  1  to  10  is  followed  by  a  second  change  of  opposite 
kind  and  of  like  amount  ? 

12.  What  is  the  total  result  of  two  transactions,  one  giving 
a  gain  of  %  50,  and  one  a  loss  of  $  25  ? 

What  single  change  will  produce  the  same  result  as  the  two 
changes  indicated  in  the  following  examples  ? 

13.  If  a  ship  sails  first  6°  north  and  then  2°  south  ? 

14.  If  a  ship  sails  first  8°  east  and  then  10°  west  ? 

15.  If  a  boy,  becoming  ill,  loses  10  pounds,  and  then  gains 
8  pounds  ? 

16.  If  the  temperature  first  rises  12°,  and  then  falls  15°  ? 

17.  If  a  man  first  deposits  $  100  in  a  bank  and  then  with- 
draws $  125  from  his  account  ? 

18.  A  vessel  sails  from  the  equator  due  north  28°,  and  then 
due  south  57°.    W^hat  is  her  latitude  at  the  end  of  the  voyage  ? 

20.  In  Example  10,  Exercise  11,  the  opposite  of  adding  7  is 
subtracting  7.  Addition  is  always  indicated  by  the  sign  +, 
and  subtraction  by  the  sign  — .  These  same  signs  are  used  to 
distinguish  between  opposite  quantities.  A  quantity  pre- 
ceded by  the  +  sign  is  called  a  Positive  Quantity,  and  one 
preceded  by  the  —  sign  is  called  a  Negative  Quantity.  Quan- 
tities preceded  by  the  signs  plus  and  minus  are  called  Signed 
Quantities. 

EXERCISE  12 

The  following  are  naturally  considered  positive  quantities. 
What  are  the  corresponding  negative  quantities  ? 

1.  Sailing  east.  5.  Eising  temperature. 

2.  Sailing  north.  6.  Forward. 

3.  Right  direction.  7.  Upward. 

4.  Increasing.  8.  Deposits. 


POSITIVE  AND  NEGATIVE   NUMBERS 


25 


9.  Assets.      10.  Prolits.      11.  Above  zero.      12.  Time  a.d. 


fS 


c 

t 

> 

100 

- 

90 

- 

ec 

- 

: 

70 

- 

60 
50 

~ 

40 

- 

30 
20 
10- 
0 

:j 

10 

20 

:_  - 

ao 

i     - 

40 

< 

^  " 

U 

13.  At  7  A.M.  the  temperature  is  —  13° ;  at  noon 
it  is  8°  warmer,  and  at  6  p.m.  it  is  5°  colder  than 
at  noon.  Required  the  temperature  at  noon  and 
at  6  P.M. 

14.  At  7  A.M.  the  temperature  is  +0°;  at  noon 
it  is  14°  colder,  and  at  6  p.m.  it  is  2°  colder  than  at 
noon.     Required  the  temperature  at  noon   and  at 

6  P.M. 

15.  At  7  A.M.  the  temperature  is  —  7°,  and  at 
noon  -t-  9°.  How  many  degrees  warmer  is  it  at 
noon  than  at  7  a.m.? 

16.  The  temperature  at  6  a.m.  is  —  14°;  during 
the  morning  it  grows  warmer  at  the  rate  of  3°  per 
hour.     Required  the  temperature  at  9  a.m.  and  at 

10  A.M. 

17.  The  positive  quantities  in  this  set  indicate  rise  in  tem- 
perature.    What  single  change  will  produce  the  same  result  ? 

(a)   -f  12°  and  + 10°. 

Solution  :  1.   A  12°  rise  followed  by  a  10'  rise  gives  a  total  of  22°  rise. 
2.    This  may  also  be  expressed  thus  : 

(+12°)  +  (+  10°)  =  +22°. 
(b)   +  9°  and  -  5°.       (c)   -  4°  and  -  5°.       (d)   +  7°  and  -  9°. 

18.  In  this  set  the  positive  quantities  refer  to  gains  in  finan- 
cial transactions.     What  is  the  equivalent  single  change? 

(a)   +$15,    -f  !i>25,    -$30. 

Solution  :  1.   A  gain  of  $15  followed  by  a  gain  of  §25  gives  a  gain  of 
$40  ;  followed  by  a  loss  of  $30  gives  a  total  result  of  $  10  gain. 
2.    Expressed  in  symbols  thus : 

(+.S15)  +  (+326)  +  (-$.30)  =  (+$40)  +  (-$30)  =  +  $10. 

(b)    4-  $20,  +  $30,  -  $50.         (c)  -  $45,  +  $50,  -  $10. 
(d)  +  $35,  -  $20,  -  $25.  (e)  -h  $14,  -  $20,  +  $19. 


26  ALGEBRA 

In  the  following  problems,  select  and  mark  the  positive  and 
negative  quantities.  Find  the  total  result  as  in  the  preceding 
problems  and  express  it  as  a  signed  number. 

19.  A  man's  income  during  the  year  is  $  1500,  and  his  ex- 
penses are  $1300.     Find  the  result  at  the  end  of  the  year. 

20.  A  man's  monthly  account  book  shows  the  items :  salary 
$150,  rent  $40,  food  $50,  insurance  $25,  interest  on  savings 
$  15.     Find  the  result  at  the  end  of  the  month. 

21.  Positive  and  Negative  Numbers.  The  preceding  exercises 
show  that  positive  and  negative  quantities  exist.  In  dealing 
with  these  quantities,  positive  and  negative  numbers  are 
necessary. 

Starting  with  an  arithmetical  number  like  3,  a  new  number 
called  Negative  3  is  made ;  3  is  then  called  Positive  3.  These 
two  numbers  are  opposites  and  have  the  power  of  destroying 
each  other  when  added,  just  as  do  opposite  quantities  which 
are  equal  in  amount.  Positive  3  is  written,  -+-  3 ;  negative  3  is 
written,  —  3.  The  arithmetical  number  3  is  called  the  Absolute 
Value  of  +  3  and  -  3. 

The  negative  number  must  always  have  its  sign  written  before  it ;  the 
positive  number  is  often  written  without  its  sign. 

Historical  Note.  — Hindu  mathematicians,  who  knew  about  positive 
and  negative  numbers  long  before  European  mathematicians,  in  referring 
to  them,  used  words  which  correspond  to  our  words  for  assets  and  debits. 
They  also  were  acquainted  with  the  illustration  of  such  numbers  by  means 
of  the  opposite  directions  on  a  straight  line.  To  indicate  that  a  number 
was  a  negative  number,  they  placed  a  dot  over  it.  European  mathemati- 
cians did  not  arrive  at  an  equal  understanding  of  positive  and  negative 
numbers  until  the  sixteenth  century. 

EXERCISE  13 

1.  Kead  the  following  numbers,  write  each  in  symbols,  and 
tell  the  absolute  value  of  each : 


POSITIVE   AND  NEGATIVE  NUMBERS  27 

(a)  positive  nine ;  (b)    negative  seven ; 

(c)   positive  four ;  (d)    negative  three  fourths ; 

(e)    negative  five  sixths ; 

(/)  negative  three  and  three  tenths. 

2.   Kead  the  following  numbers  and  tell  the  absolute  value 
of  each : 

(a)    +6;  (6)    -2;  (c)     -5; 


(d)  -1; 

W  +i; 

(./•) 

-2.98; 

(y)    +3.41; 

(7i)   -45.087; 

(0 

- 102.34, 

22.  Addition  of  Positive  and  Negative  Numbers.  The  rules  for 
addition  of  signed  numbers  are  suggested  by  the  following 
problems : 

1.  Find  the  sum  of  -f  5  and  -|-  3. 

Just  as  $5  gain  plus  $3  gain  gives  $8  gain,  similarly 
(+5)+(+3)  =  +  8. 

2.  Find  the  sum  of  —  5  and  —  3. 

Just  as  §5  loss  plus  $3  loss  gives  $8  loss,  similarly 
(_5)  +  (-3)  =  -8. 

3.  Find  the  sum  of  -h  5  and  —  3. 

Just  as  §5  gain  plus  §3  loss  gives  .$2  gain,  similarly 
(+6)  +  (-8)  =  +2. 

4.  Find  the  sum  of  —  5  and  +  3. 

Just  as  §5  loss  plus  $3  gain  gives  $  2  loss,  similarly 
(-6)  +  (+3)  =  -2. 

Rule.  —  1.  To  add  two  positive  numbers,  add  their  absolute  values 
(§  21),  and  prefix  the  plus  sign  to  the  result. 

2.  To  add  two  negative  numbers,  add  their  absolute  values  and 
prefix  the  minus  sign  to  the  result. 

3.  To  add  a  positive  and  a  negative  number,  find  the  difference  of 
their  absolute  values,  and  prefix  to  the  result  the  sign  of  the  number 
having  the  greater  absolute  value. 


,28  ALGEBRA 

EXERCISE  14 

1.  Find  the  sum  of  -f  10  and  —  3. 

Solution  :  Use  Rule  3.     Subtract  3  from  10  ;  prefix  +  sign. 
(+10)  +  (-3)  =  +7. 

2.  Find  the  sum  of  - 12  and  +  6. 

Solution  :  Use  Rule  3.     Subtract  6  from  12  ;  prefix  —  sign. 

(-12)  +  (+6)=-6. 

3.  Find  the  sum  of  —  9  and  —  5. 
Solution  :  Use  Rule  2.     Add  9  and  5  ;  prefix  —  sign. 

(_9)  +  (-5)  =  -14. 

4.  To  each  of  the  numbers : 

+  6,    +9,    +14,    +5,    +18,    +3, 
(a)  add  -4;         (b)  add  -12;         (c)  add- 15. 

5.  To  each  of  the  numbers : 

-5,    -15,    -2,    -10,    -16,    -20, 
(a)  add  +  6  ;         (b)  add  +  15 ;         (c)    add  +  12 ; 
(d)  add  -5;         (e)  add  -6;  (/)  add  -7. 

6.  To  each  of  the  numbers : 

_36,    +48,    -17,    -25,  +24,    +29, 
(a)  add  +  9 ;         (b)  add  -  8 ;         (c)  add  -  7. 


Find  the  sum : 

7.          8. 

9. 

10. 

11. 

+  124       - 15.2 

-1.35 

+  2.10 

-10.3 

-36        +  9.1 

- 1.63 

-1.43 

+  3.3 

12.    +    9  Here  9  must  be  taken  from  17.     It  is  necessary  to  be- 

—  17     come  expert  in  subtracting  the  upper  number  from  the 


- 

_  g  lower. 

13. 

14. 

15. 

16. 

17. 

18. 

-11 

+  17 

-2S 

+  72 

+  65 

-59 

+  43 

-42 

+  44 

-109 

-247 

+  78 

POSITIVE 

;   AND 

NEGATIVE 

NUMBERS 

29 

19. 

+  0 

Hint: 

There  are  two  ' 

ways  of  doing  this  example. 

Either 

-2 

add  in  oi 

•der  fron 

I  bottom  to  toi 

),  or  from  top  to  b 

ottom  ; 

+  4 

or,  add  first  all  of  the  positive  nuinbei-s  and  idl  of  the  nega- 

— 5 

tive  numbers  separately, 

and  then  combine  the  results. 

20. 

21. 

22. 

23. 

24. 

25. 

f-4 

-7 

+  15 

-   5 

+  16 

-10 

-6 

-5 

-12 

-14 

-11 

+  13 

-2 

4-9 

+   6 

+  11 

—   7 

-14 

f-5 

-1 

+   6 

+  3 

+  22 

+   7 

-3 

+  8 

-   9 

-   8 

-   9 

-   3 

Find  the 

• 
sum: 

26. 

-i 

+i- 

28. 

-h 

-h 

30.    +2^, 

-If. 

27. 

-i 

9 

29. 

-h 

+  f 

31.    -3i, 

+  2|. 

32.  Demosthenes  was  born  in  the  year  —  385,  and  died  at 
the  age  of  63.     What  was  the  year  of  his  death? 

33.  Pythagoras  was  born  in  the  year  —  580,  and  lived  to  the 
age  of  79  years.     What  was  the  year  of  his  death  ? 

34.  At  the  beginning  of  the  month  the  number  of  pupils  in 
a  schoolroom  is  53;  3  pupils  enter  during  the  month  and  5 
leave  school.  How  many  pupils  are  there  in  the  room  at  the 
end  of  the  month? 

35.  A  principal  of  a  school  finds  that  his  six  algebra  classes 
have  the  enrollment  indicated  in  line  2.  He  decides  to  make 
the  changes  indicated  in  line  3. 

Classes  12  3 

Members         29  15  22 

Changes        —  8        +6         —  1 

(a)  Tell  what  each  of  the  changes  in  line  3  means. 

(b)  Find  the  result  in  the  class  membership. 

(c)  How  can  he  check  his  work  from  line  3  ? 


4 

5 

6 

28 

18 

14 

-7 

+  3 

+  7 

30  ALGEBRA 

23.  From  the  examples  in  Exercise  14  it  is  clear  that  adding 
sL  negative  number  has  the  same  effect  as  subtracting  the  posi- 
tive number  of  equal  absolute  value. 

Thus,  (+12)  +  (-,5)=12-5  =  7. 

( _  10)  +  (_  5)  =  -  10  -  5  =  _  15. 

24.  It  is  convenient  to  picture  the  positive  numbers  thus : 

0        +1       +2        +3       -^4      -t-5 


where  -f  1  is  placed  at  any  distance  from  the  point  A,  -{-  2 
twice  as  far,  etc.  In  this  sense,  the  positive  ntimbers  form  a 
scale  extending  to  the  right.  Any  number  precedes  all  larger 
numbers  on  this  scale ;  thus,  3  precedes  4,  5,  6,  etc. 

Since  (— 3)-i-(+3)=  0,  it  is  natural  to  think  of  —  3  as 
being  3  less  than  zero ;  and,  similarly,  of  —  4  as  being  4  less 
than  zero. 

It  is  natural  to  think  of  the  negative  numbers  as  arranged 
on  the  left. 

-5      -4       -3       -2       -1           0       -fl      4-2       -1-3      -f-4      -4-5 
< • • • ■ • • • • • • • >- 

A 

Thus,  the  positive  and  negative  numbers  together  form  a 
complete  scale  extending  in  both  directions  from  zero. 

Starting  on  the  left,  any  negative  number  precedes  the  posi- 
tive numbers  and  may  he  thought  of  as  being  less  than  the  posi- 
tive yiumhers. 

MULTIPLICATION  OF  POSITIVE  AND  NEGATIVE  NUMBERS 

25.  The  terms  Multiplier,  Multiplicand,  and  Product  have  the 
same  meaning  in  algebra  as  in  arithmetic. 

The  rules  for  multiplication  of  signed  numbers  are  sug- 
gested by  the  following  problems.  In  these  problems,  read 
the  sign,  x ,  "  multiplied  by." 


POSITIVE   AND  NEGATIVE  NUMBERS  31 

1.  Find  the  product  of  +4  and  +3. 

Since  positive  numbers  are  like  arithmetical  numbers, 

(+4)  X  (+3;  =  +  12. 

2.  Find  the  product  of  —  4  and  4-  3. 

In  arithmetic,  to  multiply  by  3  means  to  add  the  multiplicand  three 
times.     If  this  is  done  in  this  problem, 

(-  4)  X  (+  3)  =  (-  4)  +  (-  4)  +  (-  4)  =  -  12. 

3.  Find  the  product  of  -f-  4  and  —  3. 

In  arithmetic,  4x3  =  3x4,  since  each  equals  12.  The  order  of  the 
factors  may  be  changed.  If  it  is  assumed  that  the  same  law  holds  in 
algebra, 

(-f  4)  X  (-3)  should  equal  (—  3)  x  (+  4)  or,  —  12,  by  problem  2. 

Then,  (+4)  x  (-3)  =  -12. 

To  multiply  a  number  by  a  negative  number  seems  to  be  accom- 
plished by  multiplying  it  by  the  absolute  value  of  the  multiplier 
and  changing  the  sign  of  the  product. 

4.  Find  the  product  of  —  4  and  —  3. 

The  multiplier  is  a  negative  number.  If  the  suggestion  from  problem 
3  is  followed,  the  result  in  this  problem  may  be  obtained  by  multiplying 
(—4)  by  3  and  changing  the  sign  of  the  product. 

(—4)  X  3  =  —  12.     Changing  the  sign  of  —  12  gives  +  12. 

Therefore,  (_  4)  x  (-  3)  =  +  12. 

Gathering  together  the  results  of  problems  1,  2,  3,  and  4 : 

1.  (+4)x(+3)=+12.  3.    (+4)x(-3)  =  -12. 

2.  (-4)  X  (4-3)  = -12.  4.    (-4)  X  (-3)  =  4-12. 

Rule.  —  To  multiply  one  signed  number  by  another : 

1.  Find  the  product  of  their  absolute  values.    (See  all  four  above.) 

2.  Make  the  product  positive  if  the  multiplicand  and  multiplier 
have  like  signs.     (See  1  and  4  above.) 

3.  Make  the  product  negative  if  the  multiplicand  and  multiplier 
have  unlike  signs.     (See  2  and  3  above.) 


S2  ALGEBRA 

EXERCISE  15 

1.  Multiply  the  numbers  : 

+  5,    +6,    +8,    +7,    +9,    +12, 
(a)  by  +4;     (h)  by  -3j     (c)   -8j     (d)  by  -9, 

2.  Multiply  the  numbers  : 

-6,    -10,    -8,    -7,    -15,    -12, 
(a)  by  +4;     (b)  by  -9;     (c)  by  -6;     (fZ)  by  +7. 

3.  Multiply  the  numbers : 

+  11,    -9,    +14,    +12,    -25,    -15, 
(a)  by  -6;     (?>)  by  +7;     (c)  by  -8.;     (d)  by  -10. 

Find  the  products  of  the  following  factors : 

4.  -9,-11.  8.    -13,  +8.  12.    +1,--^. 

5.  -7,  +12.  9.    -9,  +12.  13.    -h+h 

6.  +9,-20.  10.    -11,-20.  14.    -|,  -f 

7.  +6,  -13.  11.    +9,  -16. 
15-   Find  the  product  of  -  2,  +  3,  -  4. 

(_2).(+3)  =  -6;  (_6).(-4)  =  +24. 

16.  -  1,  +  3,  -  2.  20.    -6,-3,-4. 

17.  +3,-4,-5.  21.    -7,  +2,  -5. 

18.  -6,-2,-5.  22.    -  9,  +5,-4. 

19.  +5,-8,-3.  23.    +  10,  -  7,  +6. 

24.  If  the  number  of  negative  factors  is  even,  what  is  the 
sign  of  the  product  ? 

25.  If  the  number  of  negative  factors  is  odd,  what  is  the 
sign  of  the  product  ? 

26.  Powers  of  Positive  and  Negative  Numbers. 

(+2)2  =  (+2)(+2)  =  +  4.  (§15) 

(+2)^=(+2)(+2)(+2)  =  +  8. 
(+2y  =  (+2)(+2)(+2)(+2)=  +  16. 
It  is  clear  that  every  power  of  a  positive  number  is  positive. 


POSITIVE  AND  NEGATIVE  NUMBERS  33 

(-2/=(-2)(-2)  =  +  4. 
(-2)'  =  (-2)(-2)(-2)  =  -8. 
(-2)^  =  (.-2)(-2)(-2)(-2)  =  +  16. 
(-2)==(-2)(-2)(-2X-2)(-2)=-32. 
It  is  clear  that  every  even  power  of  a  negative  number  is 
positive  and  every  odd  power  is  negative. 

EXERCISE  16 

Find  the  values  of  the  following  powers : 

1.  (-3)2.                     5.    (-5)3.  9.  (-3)^ 

2.  (-3)'.                     6.  •(-2)«.  10.  (-5)2. 

3.  (+Af.                      7.    (-4)\  11.  (-0)3. 

4.  (-1)^                      8.    (+1)'.  12.  (-1)'. 

13.  Find  the  value  of  ax^  -\-bx-^c 

when  a  =  1,  6  =  2,  c  =  —  3,  x  =  —  2. 

ax«  +  6x  +  c  =  1  .  (-  2)2  +  (-f  2)(-  2)  +  (-  3) 
=  +4-4-3=-3. 

In  the  following  examples,  let  a  =  + 1,  6  =  —  2,  c  =  +  3,  and 
x  =  —  2.     Find  the  values  of  the  expressions : 

14.  Sab.  18.    bc-\-ax.  22.    ab'^-\-bc\ 

15.  a52.  19.    a26  +  a62.  23.    ^^.^a.-^. 

16.  5  6x2.  20.    bx^-cx^  24.    a*  -  a;*. 

17.  — 11  6ca;.  21.   a:f^—ba^.  25.    a^**  — a'c. 


III.    ADDITION   AND   SUBTRACTION   OF   ALGEBRAIC 
EXPRESSIONS 

DEFINITIONS 

27.  A  Monomial  or  Term  is  an  expression  whose  parts  are 
not  separated  by  the  signs  +  or  — .     Thus, 

2ic2,  —  3a6,  and  -f-  5  are  the  terms  of  the  expression  2  a;^  —  3  a6  +  5. 

In  an  expression,  a  term  whose  sign  is  plus  may  be  called  a  positive 
term,  and  one  whose  sign  is  minus,  a  negative  term,  as  the  terms  2  a-'-,  and 
—  3  ah,  respectively,  although  the  algebraic  value  of  the  term  depends 
upon  the  values  of  its  literal  factors. 

28.  If  two  or  more  numbers  are  multiplied  together,  each  of 
them,  or  the  product  of  any  number  of  them,  is  called  a  Factor 
Of  the  product. 

Thus,  a,  h,  c,  ah,  ac,  and  6c  are  factors  of  the  product  ahc, 

29.  Any  factor  of  the  product  is  called  the  Coefficient  of  the 
product  of  the  remaining  factors. 

Thus,  in  2  ah,  2  is  the  coefficient  of  ah,  2  a  ot  h,  a  of  2  6,  etc. 

30.  If  one  factor  of  a  product  is  expressed  in  numerals  and 
the  other  factor  is  expressed  in  letters,  the  former  is  called  the 
Numerical  Coefficient  of  the  latter. 

Thus,  in  2  ah,  2  is  the  numerical  coefficient  of  ah. 
If  no  numerical  coefficient  is  expressed,  the  coefficient  1  is  understood  ; 
thus,  a  is  the  same  as  1  a. 

31.  By  §  25,  (—  3)  X  a  =  —  3  a ;  that  is,  —  3  a  is  the  product 
of  —3  and  a.  Then  —  3  is  the  numerical  coefficient  of  a  in 
-3  a. 

Thus,  in  a  negative  term,  the  numerical  coefficient  includes 
the  sign. 

34 


ADDITION   AND  SUBTRACTION  85 

32.  Terms  wliich  are  alike  in  their  literal  parts  are  called 
Like  or  Similar  Terms ;  as,  2  xhj  and  —Ix^y. 

Terms  which  are  not  alike  in  their  literal  parts  are  called 
Unlike  or  Dissimilar  Terms ;  as,  2  ar^  and  7  xif. 

Sometimes  unlike  terms  may  be  like  with  respect  to  one 
or  more  letters.  Thus,  2  axy  and  3  hxy  are  like  with  respect 
to  xy. 

EXERCISE  17 

1.  Tell  all  of  the  factors  of  the  monomial  4-  6  xy. 

2.  What  is  the  numerical  coefficient  of  the  monomial  —&xy? 

3.  In  2,  what  is  the  coefficient  of  y  ?  of  a; ?  of  xy  ?  oi  6  x? 

4.  Select  the  sets  of  like  terms  : 

(a)   -\-2xy^,   -3xy-.  (b)    +  7  ?w,   -6n. 

(c)    4-  5  abcj  —  6  ab'^c.  {d)  —Sab,  +9  ab. 

33.  The  result  obtained  by  adding  two  or  more  numbers  is 
called  the  Sum. 

ADDITION  OF  MONOMIALS 

34.  Addition  of  Like  Terms.  In  paragraph  11,  it  was  found 
that  6  ?i  +  4  7i  =  10  n.  G  n  and  4  n  are  like  terms  since  they 
have  the  common  factor  n  (§  11).  The  coefficients  of  ii  are  6 
and  4,  and  their  sum  is  the  coefficient,  10,  of  the  result. 

Rule.  —  To  add  two  or  more  like  terms : 

1.  Multiply  their  common  factor  by  the  sum  of  its  coefficients. 

Example  1.    Find  the  sum  of  5  a:^  and  3  x^. 
Solution:  1.    The  common  factor  in  hx^  and  Sx^isx^,     The  coeffi- 
cients are  5  and  3  ;  their  sum  is  8. 

2.  Hence,  6  a-2  +  3  j2  =  g  x^. 

Check  :  Let  re  =-  2.  Sx*^  =  5  •  (-  2)2  =  5  •  4  =  20,  3  x^  =  3  .  4  =  12, 
and  20  +  12  =  32.     Also,  8  x^  =  8  •  4  =  32. 

Example  2.     Find  the  sum  of  —  5  xhj  and  +  3  x^y. 
Solution  :  -  5  x^y  +  3  x^y  =  [(-  5)  +  (+  3)]  x^y  =_  2  x^ 


36  ALGEBRA 

Check:  Let  z  =  l  and  y  =  1.  Then,  —  5x^  =—  b  ■  1  •!=—  5, 
+  3a;2y  =  +  3.  1  .  l=+3,  and  (_  5)  +  (+ 3)  =  - 2.  Also,  -2x'h/  = 
C-2).l.l=-2. 

Example  3.     Find  the  sum  of  16  x  19  and  14  x  19. 
Solution  :  1.   The  common  factor  is  19;  its  coefficients  are  16  and  14. 
2.    16  X  19  +  14  X  19  =  (16  +  14)  x  19  =  30  x  19  =  570. 

EXERCISE   18 
Find  tlie  sum  in  each  of  the  following : 

1.  5  A  and  -  12  A.  8.    - 17  a^  and  -  15  x". 

2.  lip  and  —6 p.  9.   xyz  and  —  9  xyz. 

3.  -  7  m  and  -  8  m.  10.    8  xY  and  -  29  a^/. 

4.  —4n  and— 9  72.  11.    —  &cand-f6  6c. 

5.  + 15  J5:  and  -  11  jE;.  12.   12  x  16  and  8  x  16. 

6.  -5ab  and  +  13  ab.  13.   21  x  17  and  9  x  17. 

7.  -  13  r^s  and  +  36  A  14.    13  x  23  and  7  x  23. 


15. 

16. 

17. 

18. 

19. 

-f-7a 
4-9a 

4-  15a^ 
-    4:xy 

-2xf 
-9xf 

4-    5m 
—  15  m 

20. 

-7a 

4-9a 
+  8a 
-3a 

Hint.     Add  first  the  positive  terms,  getting  +  17  a  ;  then 
the  negative,  getting  —  10  a  ;  then  add  these  results,  getting 
+  7  a.     All  should  be  done  mentally. 

+  7a 

21. 

22. 

23. 

24. 

•    25. 

15  m 

-2m 
—  5m 
+  3m 

-ISx" 

+    2x' 
-    Ix" 
+    3ar' 

+  8a5 
-7a5 
-5o1j 
+  3  ab 

16  xyz 

-4  xyz 
—  6  xyz 
+  3  xyz 

21  ab'C 

-  6abh 

-  17  ab'c 
+       ab^c 

ADDITION  AND  SUBTRACTION  91 

26.    Find  the  sum  of  9  a'',  —  7  a^,  and  +8  a^ 

Solution  :   9  a^  +  (-  7  a^)  -\-S  a^  =  9  a^ -7  a^ -\-Sa^  =  10  a^. 

Note.  This  illustrates  another  arrangement  of  an  addition  example.  The 
terms  are  first  connected  by  the  +  sign.  Then  +(—7  a^)  is  written  as  —  7  a^, 
since  adding  a  negative  number  is  the  same  as  subtracting  the  positive  num- 
ber of  equal  absolute  value.     {§  21.) 

Add,  as  in  Example  26,  the  following: 

27.  Tab,  —  3  aft,  and  +  9  a6.       29.  122/, —7  y, +9  y,  and  — 8 y. 

28.  9?-^,  -Gr',  +3?'»,andl0r».   30.  Wz,  -^Oz,  -11 2;,  and  -8z. 

31.  Find  the  sum  of  12  a,  —5x,  —  3  y^,  —5  a,  and  8  x. 
Solution  :        12  a  +  (-  5  x) -^  (-  S  y^)  +  (-  5  a) -\-(S  x) 

=  V2a'-6x-3y^-6a-\-Sx. 
=  7  a  +  3  X  -  3  y2. 

In  this  Example,  12  a  and  —5a  give  7a;  —  5  x  and  -|-  8  x  give  +  3  x  ; 
there  is  no  term  to  combine  with  —  3  y^.  Only  the  like  terms  may  be 
combined. 

Add,  as  in  Example  31,  the  following : 

32.  Sab,  —  9  cd,  —  0  ab,  and  +  4  cd. 

33.  Gx^,  -10z-,2y'',4:z%  -9y^,and  -Sx". 

34.  12r2,  -3r,  -8s,  -5r,  4-3.9,  -Tr",  and  lis. 

35.  10  c,  -  4  d,  -  3  A;,  +  9  c,  -  4  k,  +  5d,  -  6  A;,  and  -  2  c. 

36.  11  m%  -  8  ?i-,  +  6  Z,  -  5  71*,  4-  3  m^,  and  -81, 

Simplify  the  following : 

37.  5a-8a  +  10a-12a. 

38.  -3x'-\-6x'-\-x^-nx'-\-a^. 

39.  4:7^8-7  7^8-\-or^s -6  )^s. 

40.  —  8  m^n  —  3  m^n  -}-  2  m^n  —  9  m'n. 

41.  6c  + 11  c?-4ci-5c-2c. 

42.  2a?-3x-^x-5x^  +  Sx. 


88  ALGEBRA 

43.  9'i'^-\-2r''-3i^-4:r\            47.  ^xy  -  j\  xy  ~^xy. 

44..  |a  +  ia-ia.                       48.  ^f^^^f-y^-^xf. 

45.  —  \  mn  + 1  mn  +  ^  mn,         49.  2.5  ajy  —  .3  a^y  4- 1.2  a;^/- 


46.    -2,v  +  irV- irV.  50.    -  3.05  a  +  4.4  a  -  1.3  a. 


DEFINITIONS 

35.  A  Polynomial  is  an  algebraic  expression   consisting  of 
two  or  more  terms  :  as,  a  -f  6,  or  2  a^^  —  3  a;?/  +  5  y"^. 

A  Binomial  is  a  polynomial  of  two  terms  :  as,  a  +  h. 
A  Trinomial  is  a  polynomial  of  three  terms. 

ADDITION  or  POLYNOMIALS 

36.  Addition  of  polynomials  is  similar  to  addition  of   de- 
nominate numbers  in  arithmetic. 

Example  1.     Find  the  sum  of  3  yd.  2  ft.  and  6  in.    and 
5  yd.  1  ft.  and  4  in. 

Solution  :  3  yd.  +  2  ft.  +    6  in. 

5  yd.  +  1  ft.  4-    4  in. 
8  yd.  +  3  ft.  +  10  in. 

Example  2.     Find  the  sum  of  2  a+3  6+5  c  and  4  a  +  6+2c. 
Solution  :  2a  +  36  +  5c 

4a+     &  +  2c 
«a  +  46  +  7c 

Rule.  —  To  add  two  polynomials : 

1.  Rewrite  the  polynomials,  if  necessary,  so  that  like  terms  are 
in  the  same  vertical  column. 

2.  Add  the  columns  of  like  terms. 

3.  Write  the  results  of  step  2  with  their  proper  signs  for  the  sum. 

Example  1.     Find  the  sum  of  6  a-7  a;^,  3  a;^  _  2  «  +  3  2/^  and 
2x^—  a  —  mn. 

Solution  :  6  a  —  7  a;2 

-2a  +  3a:2  +  3?/2 
—    a  +  2  a;2  _  inn 

3a  — 2a;2  ^hy^  —  mn 


ADDITION  AND  SUBTRACTION  89 

Check":  Letting  a  =  2,   oc  =  1,   y  =  1,   m  =  2,    n  =  1  : 
6a-7a;2=12-7  =  5;~2a  +  3a;2  +  3y2  =  _44.3^_3  =  _^2; 

-«  +  2x'2-m7i=-2  +  2-2=-2; 
and  the  sura  of  these  values  of  the  polynomials  is  5  +  2  —  2  or  5. 
The  value  of  the  sum  of  the  polynomials, 

3  a  -  2  x2  +  3  y2  _  ,n?i  is  C  -  2  +  3  -  2  or  5. 

Since  the  two  values  are  equal,  the  solution  is  probably  correct.  This 
is  called  checking  by  substitution. 

Another  method  of  checking  is  to  add  carefully  in  the  opposite  direc- 
tion, as  in  arithmetic. 

Example  2.  Find  the  sum  of  x^-f-\-Sxf -3  oc^y,  2a^-\- 
3xtf-{-  if,  and  2  a;^?/  —  4  xif^.     Check  f or  x  =  1,  ?/  =  1. 

Solution  :  1.      a:«  -  3  ar^y  +  3  a;?/^  -  y*  =  1  -  3  +  3  -  1  =     0 

2  x8  +  3ici/'-2  +  y"  =  2         +  3  +  1  =     0 

+  2  ofiy  -  4  xy'^  =         2-4         =-2 

Sofi-    x'^+2xy^  +4 

Also,  3  a^  —  a;2y  +  2  xy2  _  3  _  1  ^  2  =  +  4.     The  solution  is  correct. 

37.  In  Example  2  of  this  last  paragraph,  notice  that  the 
three  polynomials  have  been  rearranged.  The  term  containing 
the  highest  power  of  x  is  placed  first;  then  the  term  having 
the  next  lower  power  of  x,  and  so  on.  The  term  —  f,  not  con- 
taining any  x,  is  thought  of  as  containing  the  lowest  power  of 
X  in  this  expression.  The  polynomials  are  arranged  in  Descend- 
ing Powers  of  x. 

These  polynomials  are  arranged  also  in  Ascending  Powers  of 
y.    The  exponents  of  y  in  the  terms  increase  from  left  to  right. 

In  all  examples  in  addition,  subtraction,  multiplication,  and 
division  of  polynomials,  it  is  advisable  to  arrange  the  polynomi- 
als, if  necessary,  according  to  ascending  or  descending  powers 
of  one  of  the  letters  when  writing  them  preparatory  to  solving 
the  example.  When  the  polynomials  are  so  arranged,  there 
is  less  likelihood  of  making  some  error  in  the  solution,  compari- 
son and  checking  of  results  is  facilitated,  and  the  final  solution 
has  a  more  workmanlike  and  finished  appearance. 


40  ALGEBRA 

EXERCISE  19 

Find  the  sums  in  the  followinnr : 


1. 

2 

1. 

3. 

7A-3B 

-    Qf  + 

5n^ 

—  16  am  ■+■  14  &» 

-9A-i-2B 

+  12t'- 

13  7l« 

6  am  —    5bn 

-\-SA-     B 

-  ^e  +  i^n^ 

9  a?>i  —    5  6n 

4. 

5. 

a^-2ah  + 

W 

7^ 

-    6rs^ 

a^  -f  4  a&  - 

2  b' 

+  6r's                -10 

-2a^-2ah 

-11^^ 

—  5  r-s  4- 15  rs^ 

6. 

7. 

3a?-^x'y  +  2xy'  -4.f 

—  7  m^ 

4-  5  mn^ 

■^Qo^y 

-h^f 

-  6  mhi  - 13  71171^ -\- 2  71^ 

-5a^ 

7  xy^ 

+  4m-/i+   o7)in^ 

Note.    In  Examples  5  and  7,  blank  spaces  are  left  when  the  first  poly- 
nomial is  written  for  powers  which  are  not  present  in  the  polynomial. 

8.  3a  —  46  +  6c  and  5a-]-7b  —  Sc. 

9.  4A:-7  +  3m,  5A:  +  2-4m,  and3A;-2. 

10.  2  a  +  8  6  -  5  c,  -  3  6  -  3  c  -f  7  d,  and  +  6  c  -  4  a  +  2  cZ. 

11.  12r  +  6s-9^,  8r-9s  +  ll^,  and  15r-}-7s-6t 

12.  x^  -\-  2xy  +  y%  x^  -  2  xy  -{-  y^,  and  -2  x^  -  2  y^ -^  lOxy. 

13.  4  m^  —  4  mn  +  7i^,  m^  +  4  mn  -f  4  n^,  and  —  5  m^  +  5  n^ 

14.  a^  -f  3  a26  -f  3  a&^+fts  and  a«  -  3  a^fe  +  3  aZ^^  _  ft^. 

15.  3a^-^9xy-'i7f,  -2xy~5x--10y',and7  7/-6xy-\-Sx'. 

16.  15  a^  -  6  +  4  a  4- 16  a^  and  -  2  -  13  a  +  8  a^  -  3  a^ 

17.  a^  +  a^?/^+a^2/^+a;^3/4-ic?/'*+^  and  x^—x*y-{-xy*-\-x^y^—y^. 

18.  12  a^- 4  a +  3,  -8a;4-6-f  15  a;3-f  2  a^,  and  -16  +  6a.-2 
-  8  a.-3  -  11  a;. 


ADDITION    AND  SUBTRACTION  41 

19.  9  A^  4-  3AB'  -  B",  -  2  A'B  -f-  5  AB'  +  7  2^\  and  -6A^ 
4-13  ^2^. 

20.  ia  +  lb  —  Ic  and  J-  «  —  ^  ^  +  iV  c. 

'21.  I  ?/i  —  I  n  4-  ^j  — i  *^  +  A  ^*  —  i  ^>  ^^i<i  2  m  +  y^o^  n  —  ^  r. 

22.  3a-i6-3^cand-fa-|6  +  fc. 

23.  2.3  ^1  -  6.02  5  4-  3.5  (7  and  -  1.6  A  -  4.38  B-2a 
24!  2.25  a^  -  3.5  cr2/  +  4  2/2  and  -  1.5  a^  +  2.75  a;//  -  3.2  1/. 

25.  3m—  1.3  71  +  4  and  .5  m  +  2  w  —  1.6. 

26.  2a2-5a6-6^7a2  4-3a6-9ft2,and  -4.a^-Gab +  Sb\ 

27.  4  a;  -  3  a;2  -  11  +  5  a-^  12  ar^  -  7  -  Sa:^  -  15  a;,  and  14 

+  6  ar'  +  10  a;  -  9  x". 

28.  x^  —  3  .avy^  —  2  ar^2^,  3  a;^?/  —  5  ?/^  —  4  a;t/^,  5  xi/  —  6y^  —  7  a^, 
and  8  ^  +  7  ar*  —  9  ar^y. 

29.  15a«-2-9a*-3a,13a-5a2-6-7a^8+4a-8a3-7a^ 
and  16  a^  +  3  a'  -  10  a  -  2. 

30.  12  cv'-x'  +  4  «a;2  -  5  a\  18  ar''  -  2  a^a:  -  3  a^  -  13  aa;^, 
15  a-x  +  11  ar"'  -  17  a^  +  3  ax",  and  6  aa:"  -  8  a^a;  -  7  ar^  +  9  al 

SUBTRACTION 

38.  Subtraction  is  the  process  of  finding  what  number  must 
be  added  to  one  given  number  to  produce  another  given 
number. 

Thus,  subtracting  3  from  8  determines  the  number  which  must  be 
added  to  3  to  give  8  ;  and  subtracting  a  from  b  determines  the  number 
which  must  be  added  to  a  to  give  b. 

The  number  subtracted  is  called  the  Subtrahend. 
The    luimber   from  which  the  subtrahend    is  subtracted  is 
called  the  Minuend. 

The  result  is  called  the  Remainder  or  Difference. 


42 


ALGEBRA 


EXERCISE  20 
What  must  be  added  to  the  first  number  to  give  the  second  ? 

1.  4,  12.  4.   3  a,  5  a.  7.    13  c,  21  c. 

2.  11,  15.  5.   4  m,  7  m.  8.    25  6,  30  6. 

3.  12,  18.  6.    8  m,  15  m.  9.   16  x,  22  a;. 

How  much  must  the  temperature  rise  to  change : 

10.  +  2°  into  + 10°  ?  14.    -  3°  into  +  2°  ? 

11.  4-3°  into  + 12°  ?  15.    -  6°  into  +  4°  ? 

12.  -8°  into- 3°  ?  16.    -5°  into  +7°? 

13.  -  6°  into  -  2°  ?  17.    -  2°  into  +  9°  ? 

Find  the  number  to  be  added  : 

18.  (+3)  +  ?= +8.  22.  (-4) +  ?=-}- 6. 

19.  (-3)  +  ?  =  0.  23.  (-5)  +  ?  =  +3. 

20.  (-3)4-?=+2.  24.  (_8)+?  =  -2. 

21.  (-3)  +  ?  =  +4.  25.  (-6)4-?=  +  6. 

How  much  must  the  temperature  fall  to  change : 

30.  +  8°  into  -  3°  ? 

31.  -2°  into  -8°? 

32.  -h  10°  into  -  3°  ? 

33.  0°  into  -  8°  ? 


JM 


26.  +  2°  into  0°  ? 

27.  4-2°  into  -3°? 

28.  4-  4°  into  -  6°  ? 

29.  +3°  into- 5°  ? 
Find  the  number  to  be  added 

34.  (-h3)4-?  =  0. 

35.  (+3)4-?=  -2. 

36.  (+4)  +  ?= -7. 

37.  (-h5)  +  ?  =  -6. 

38.  (+2)-f?=-5. 

39.  (4-1)4-?= -7. 


40.  (4-7)+?=  -2. 

41.  (_5)4_?  =  _7. 

42.  (_4)  +  ?=-8. 

43.  (-l)+?=-9. 

44.  (-2)  +  ?= -10. 

45.  (-3)+?  =  -12. 


ADDITION   AND  SUBTRACTION 
39.    Subtraction  of  Positive  and  Negative  Numbers. 


48 


The  rule  for  subtraction  of  signed  numbers 
is  suggested  by  the  following  problems  : 

U) 

Signs  a8 
Given  ; 

SXTB  TRACT 

{B) 

SlONS 
CUANUEU ; 

Ann 

1.    Subtract  -f-2  from  +6. 

This  means  :  (+  2)  +  ?  =  +  C.     Result,  +  4. 
Hence,            (+ 6)  -  (+ 2)  =  + 4. 

+  6 
+  2 

+  4 

+  0 
-2 

+  4 

2.    Subtract  —  2  from  -f  6. 

This  means  :  (-  2)  -1-  ?  =  -f  6.     Result,  +  8. 
Hence,             (  +  6)  -  (- 2)  =:+ 8. 

+  6 
-2 

+  8 

+  6 
+  2 
+  8 

3.    Subtract  -f  2  from  —  6. 

This  means  :  (+  2)  +  ?  =-  6.     Result,  -  8.. 
Hence,            (_6)  -  (+ 2)  =- 8. 

-6 

+  2 
-8 

-6 
-2 

-8 

4.    Subtract  —  2  from  —  6. 

This  means  :  (_  2)  +  ?  =-  6.     Result,  -  4. 
Hence,             (_  6)  -  (- 2)  =- 4. 

-6 
-2 
-4 

-6 

+  2 
-4 

In  the  column  A  on  the  right,  the  problem  is  arranged  as 
usual  for  a  subtraction  problem,  with  the  result  as  it  was 
obtained  in  the  solution  on  the  left ;  in  the  column  B  on  the 
right,  the  sign  of  the  subtrahend  has  been  changed  from  4-  to  — , 
or  from  —  to  +.  Notice  that  the  correct  result  is  then  ob- 
tained by  adding.     Hence, 

Rule.  —  To  subtract  one  number  from  another,  change  the  sign  of 
the  subtrahend  and  add  it  to  the  minuend. 

Historical  Note.— The  symbol,  — ,  like  the  symbol,  +,  first 
appeared  in  print  in  a  mathematical  book  by  Widmann  in  1489.  The 
Italian  and  French  mathematicians  of  the  same  period  used  the  symbol, 
lb,  derived  from  the  first  letter  of  the  Latin  word  minus. 


44  ALGEBRA 

EXERCISE  21 

1.  Subtract  —  G  from  — 15. 

Solution  :  —  15  Imagine  the  sign  of  the  subtrahend  changed  to 

—  6       +  and  then  add  by   the  rule  in  §  22.     Do  not 

—  9       change  the  sign  on  the  paper. 

2.  Find  the  remainders  : 

4.13      4.9      4-8      -11      -14      -16      -    8      -12. 

+   4      -5      +3      4^      -_6      Hh_3      -h_10      +   9 

3.  _29      -34      4-26     -45     +37     -19     -35     +57 
+   7      -    6      -   5     +    9     +    8     +   6     +15     +13 

•     4.   From  each  of  the  minuends  of  Example  3,  subtract  —11. 

5.  From  each  of  the  minuends  of  Example  3,  subtract  +12. 

6.  From  each  of  the  minuends  of  Example  3,  subtract  —14. 

7.  The  minimum  temperature  on  a  certain  day  in  Chicago 
was  — 14  and  the  maximum  was  —  3.  What  was  the  range 
of  temperature  ? 

8.  The  Roman  nation  lived  under  a  republican  form  of 
government  from  the  year — 509  to  the  year — 31;  for  how 
many  years  did  the  republic  last  ? 

9.  Was  subtraction  always  possible  in  arithmetic  ? 

10.  Is  subtraction  always  possible  in  algebra  ?     Why  ? 

11.  Find  the  remainders  : 

+  15a     -12a;     -    7m     + 14  ^^     _^27ri     -^2xyz 
-    6a     +18  a;     +15m     -    8  n^     +34r^     + 18  xyz 

12.  -167^5         +33a;?/2         -  26  m^         -4371^         -A5xy 
_19^.2^         -l^xy"^        -A:lm^         - 17  n^         +13^-1/ 

Subtract : 

13.  —  xy  from  -\-xy.  15.    21  abc  from  —  39  ahc. 

14.  -  15  a^  from  —  46  a\  16.    -  45  ax''  from  +  19  ax\ 


ADDITION   AND  SUBTRACTION  45 

17.  Subtract  -  31  a^b  from  -  8  a'b. 

18.  Subtract  19  7's  from  —  G  rs. 

19.  Take  8  a  from  —12  a. 

20.  From  —  3  m^  take  4  m\ 

21.  Take  19  a  from  —  23  a. 

22.  Take  - 16  rv^x  from  -  27  n^'o;. 

SUBTRACTION  OF  POLYNOMIALS 

40.  Rule.  —  To  subtract  one  polynomial  from  another : 

1.  Rewrite  the  minuend,  if  necessary,  in  descending  powers  of 
some  one  letter. 

2.  Write  below  it  the  subtrahend,  having  like  terms  in  the  same 
vertical  column. 

3.  Imagine  the  signs  of  the  terms  of  the  subtrahend  changed,  and 
add  the  resulting  terms  to  those  of  the  minuend. 

Example.     Subtract  7  ab^  -  9  a'b  +  8  6^  from  -2a'b  +  i  ab^ 
-\-5a\ 

Solution  :  1.   Arrange  according  to  descending  powers  of  a. 

Minuend  :  6  a«  -  2  0^6  +  4  ab^ 

Subtrahend:  —  9 a^?>  +  7 ah'^  +  8  h^    Change  signs  mentally;  add. 

Remainder:      b  a^  +  7  aV)  -  3  ah'^  —  8  f/^ 

Check  :  The  sum  of  the  subtrahend  and  remainder  is  the  minuend. 


EXERCISE  22 

Subtract ;  check  each  in  some  way  : 

1. 

3. 

5. 

12a'-9a-   7 

17  m  — 12/1 -f6p 

2ab-\-   5bc-Sac 

Sa^  +  Qa-^IS 

20m-lGn-5p 

—  a6H-ll  6c  —  4ac 

M.,.-^^   ^::0 

2. 

4. 

6. 

5a-Sx-{-2y 

3a  +  7  6  —  c 

ar^  +  13a;-ll 

-3a-^7x-3y 

2a-h46--c 

-Sa^-h   6x-    5 

46 

7. 
lOr^             -f6rs 

ALGEBRA 

9. 

o'^Sa'b          +26^ 
a^^Sa'b-{-ab'-    b^ 

or' 

11. 

x2  +  2a;y  +  2/' 

8. 
5x'-6x 

10. 

x'                      +2/^ 

12. 

4-  xhj  -{-xy^  +  f 
-x^y-^-xy^'-f 

13.  From  5a  —  364-4c  subtract  5a  +  3&  —  4c. 

14.  From  —  2  m^  —  4  mn  +  9  w^  subtract  Sm^  —  7  mil  -\- 14  n^ 

15.  From  ab-^bc  +  ac  subtract  ab  —  bc-\-  ac. 

16.  From  4£c3-9a;2-lla;+18  subtract  3x^-8  aj2_;1^7^_j_ 25. 

17.  From  —3y-\-Sx  —  4:Z  subtract  —  z -^  11  x  —  6 y. 

18.  From  8^  +  2^-7  C  subtract  8^- 25  +  7  O. 

19.  Subtract  2a;3- 7- 4a;- 6 a;2  from  5r^-12 +  9a;'-2  a;. 

20.  From  x^  —  2xy-{-y^  subtract  x^—2xy  —  y^. 

21.  Take76-9c-2(ifrom6a-5&  +  12c. 

22.  Take  12a5  +  4a-9  from  3a'^  +  8a2-6. 

23.  Subtract  1  +  a^  —  a  —  a^  from  3  a  —  3  a'*+ 1  —  a^ 

24.  From  10  a;^  _  21  a.-^  _  11  a;  take  - 15  ar^- 20  a; +  12. 

25.  From  17  a«  -  12  ab''  +  5  6^  take  8  a^  _  3  a^^  + 13  b\ 

26.  Take  6c-5d-96-4a  from  -10&-2c+3a-9d. 

27.  Subtract    4-3a;-x2  +  8a;3  +  10a;*    from    9-7a;  +  6a;2 
-12  a;«  +  5a;^ 

28.  From  7a-ll£i3-8  +  6a''  subtract  16a2-9  +  2rt^  +  15a 

-  10  a\ 

29.  From  ar' + 3 xSj - ^y^ ■^5x^f  —  4:xy*  subtract  8 x*y  —  7 x^y^ 

-  6  xY-^  11  xy'-y\ 

30.  Subtract 

7^<,2_5_20w3  +  13ii  from  -9 -Un^  +  lGn  +  Sn^, 


ADDITION  AND  SUBTRACTION  47 

31.  Subtract 

- x^  +  '6 x^y -3x1/ -\-  f  from  ar' - 2 a^// - 2 x^-  +  f- 

32.  Subtract  -  2  x"  -  I'd -{■  ^1  a?  from  x  +  l^x"-  18. 

33.  Subtract  .V tn  —  \n  +  ^p  from  f  7rt  +  J ?i  +  f\i>. 

34.  Subtract  fa— iV^  +  A^^  ^^'^"^  2a  —  ^6  —  fc. 

35.  Subtract  -  |  v  +  i  w  —  f  «  from  fV  ■"  -f  |  «^  —  A  ^*-  * 

36.  By  how  much  does  81  ^-  +  4a2  — 36a6  exceed   —  30a6 

37.  By  how  much  does  — 5c  +  12a  — 8  6  exceed  7  a— 9  c— 6? 

38.  By  how  much  does  0  exceed  — 3a  +  26  —  c? 

39.  By  how  much  does  1  exceed  — 6a  —  46  +  6? 

40.  From  a^  —  2  a6  H-  6-  subtract  the  sum  of  —  a^  +  2  a6  —  6^ 
ami  -2a2  +  262. 

Hint  :  The  last  two  expressions  are  both  to  be  subtracted  ;  they  are  there- 
fore subtrahends  and  should  have  their  signs  changed.  Write  down  the  min- 
uend, and,  below  it,  the  two  subtrahends  with  their  signs  changed  ;  then  add, 
thus  doing  the  whole  example  at  one  operation. 

41.  From  the  sum  of  3a2-2a6  +  6"  and  ha^ -^ab  +  Qh'^ 
take  6  a^  -  4  a6  -  3  6*-. 

42.  From  9  ar*  —  8  ic  -f-  ar'  take  the  sum  of  5  —  ar^  -|-  a;  and  6  a^ 
-7a;-4. 

43.  From  the  sum  of  a;-fy  — 7  2  and  -f-4a;  — 9y  take  the 
sum  of  9 X  —  2y-\-z  and  — 5a;  +  6?/  —  7z. 

44.  From  the  sum  of  2a-f36  —  4(Z  and  26  —  4c  +  3d  take 
the  sum  of4a  —  46  —  3c-f-2d  and  3 a  +  2 c. 

45.  Subtract  .5  aj  +  .25  y  — 1.2  2  from  3  a;  — 1.75  y  -f  .8  2. 

41.  Addition  and  Subtraction  used  in  Equations.  There  are 
two  more  important  rules  used  in  solving  equations.  They 
will  be  illustrated  by  the  scales.     Review  §§9  and  10. 


48  ALGEBRA 

Suppose  that  the  sugar  S  exactly  balances  the  weight  W  on  the  scales 
in  the  figure.     If  a  3-lb.  weight  be  added  to  the  right  scalepan,  and  3  lb. 
of  sugar  to  the  left  scalepan,  then  the  scales  will 
r^      still  balance. 

, A       '^^^— 1         Similarly,  if  anv  number  of  pounds  of  weight  be 

removed  from   the  right  scalepan  and   an   equal 
weight  of  sugar  from  the  left  scalepan,  then  the  scales  will  still  balance. 

These  facts  illustrate  the  rules  : 

Rule.  —  1.  The  same  number  may  be  added  to  both  members  of 
an  equation  without  destroying  the  equality. 

2.  The  same  number  may  be  subtracted  from  both  members  of 
an  equation  without  destroying  the  equality. 

Example  1.     Solve  the  equation  a.-  —  3  =  7. 

Solution  :  1.  ic  —  3  is  3  less  than  x\  if  3  is  added  to  aj  ~  3,  the  sum  is 
therefore  x.  Add  3  to  both  members  of  the  equation  in  order  to  keep 
them  equal. 

2.  Adding  3  to  both  members,  x  —  3+3  =  7+3, 

3.  or  X  =  10. 
Check  :  Does  10  -  3  =  7  ?    Yes. 

Example  2.     Solve  the  equation  18  a;  —  5  =  3  a?  +  55. 
Solution  :  1.  18  x  —  5  =  3  a;  +  55. 

2.  Adding  5  to  both  members  of  the  equation, 

18  X  =  3  X  +  60. 

3.  Subtracting  3  x  from  both  members  of  the  equation, 

15  X  =  60. 

4.  Dividing  both  members  of  the  equation  by  15, 

X  =  4. 
Check  :    Substitute  4  in  equation  1 ;  does  18  x4  —  5  =  3x4  + 56? 
does  72- 5  =  12 +  55?    Yes. 

42.  In  order  to  abbreviate  the  explanation  of  the  solutions 
of  equations,  symbols  A,  S,  M,  and  D  will  be  used. 

Thus:  A3  will  mean  "add  3  to  both  members  of  the  equation." 
S2„  will  mean  "  subtract   2  7i   from    both  members   of  the 
equation." 


ADDITION   AND  SUBTRACTION  49 

M_3  will  mean  "  multiply   both    members    of   the  equation 
by  -  3." 

Dj  will  mean  "  divide  bot^i  members  of  the  equation  by  7." 
These  symbols  will  be  used  in  the  text  from  now  on. 
Pupils  will  find  them  helpful  when  solving  equations. 

EXERCISE  23 
Tell  what  the  following  symbols  mean: 


1.   A^. 

3. 

M,,.           5.    D.4. 

7. 

A^. 

9. 

Sz. 

2.    S_s. 

4. 

A_8,.          6.    M_i. 

8. 

D34. 

10. 

M.,. 

11.   Solve  the  equation  24  —  11 7?i  =  6 

— 

8  m. 

Solution  : 

1. 

24-llm  =  0-8m. 

2.    A8«: 

24  -  3  m  =  6. 

(Rule  1, 

,§41) 

3.    S84: 

-3m=-18. 

(Rule  2, 

,§41) 

4.    M.i: 

+  3  wi  =  +  18. 

(Rule, 

§10) 

5.    Ds: 

m  =  6. 

(Rule, 

§    9) 

Check  :  Substitute  6  in  the  given  equation. 

Does  24-11  x6  =  6-8x6?  does  24  -  66  =  6  -  48  ?  does  -42 
=  -42?     Yes. 

Note  1.  In  step  2,  "  Agm  "  means  "  add  8  m  to  both  members  of  the  pre- 
vious equation"  ;  the  result  will  be  equation  2.  In  step  4,  "M-i"  means 
"multiply  both  members  of  the  previous  equation  by  —1";  the  result  will 
be  equation  4. 

Note  2.  Whenever  the  coefficient  of  the  unknown  is  negative,  as  in  step 
3,  multiply  both  members  by  —  1  so  as  to  make  it  positive. 

Solve  the  following  equations  and  test  the  result : 

12.  a;-f3  =  12.  19.  Sk  =  S-\-k. 

13.  ?/-f7  =  15.  20.  6a;  =  ic-f-45. 

14.  m-2  =  9.  21.  7 y  =  ^y-\~  12. 

15.  2m+5  =  ll.  22.  6  2  =  63-3  2. 

16.  3aH-7=:19.  23.  10r=80-6r. 

17.  5c-2  =  23.  24.  2s  =  99-9s. 

18.  4^-7  =  21.  25.  3a  =  120-7a. 


50  ALGEBRA 

26.  15a;  =  45-15a;  34.  -  2  x -\- 5  =  -15.  . 

27.  4r  +  5  =  6  +  3r.  35.  -5y  +  9  =  ~21. 

28.  3a;  +  9  =  37-a;.  36.  -11^  +  10  =  4-82. 

29.  5^-8  =  28  +  2^  37.  16r-l  =  4r+5. 

30.  Gw-ll  =  88  — 3t^.  38.  17t-7  =  l-7t. 

31.  15A;-13  =  9A;  +  17.  39.  llm  + 6  =  -  9?/i  +  18. 

32.  46  +  15  =  35-6.  40.  13^;- 9  =  -  2^>  +  36. 

33.  3c-6  =  c  +  14.  41.  19r  +  ll=7r  +  13. 

43.  In  order  to  solve  problems,  it  is  necessary  to  translate 
the  statements  which  give  the  conditions  of  the  problem  into 
algebraic  symbols. 

Example.  One  number  exceeds  another  number  by  18. 
The  product  of  the  smaller  number  and  3  equals  the  larger 
number.     Find  the  numbers. 

Solution  :     1.  Let  s  =  the  smaller  number. 

2.  Then  s  +  18  =  the  larger  number, 

3.  and  '6  s  =  the  product  of  the  smaller  and  3. 

4.  .*.  3  s  =  s  +  18,  since  the  product  must  equal  the 

larger  number. 

5.  S,:  2s  =  18. 

6.  Dg:  s  =  9. 

Check  :     The  smaher  is  9  ;  the  larger  27  ;  and  3  x  9  =  27. 
Note.    The  symbol,  .'.,  means  "  therefore." 

Rule.  —  To  solve  a  problem  by  means  of  an  equation : 

1.  Represent  one  of  the  unknown  numbers  by  some  letter. 

2.  Represent  the  other  unknown  numbers  by  means  of  this  same 
letter,  using  relations  given  in  the  problem. 

3.  From  the  conditions  of  the  problem  form  an  equation  between 
the  numbers;  solve  the  equation. 

4.  Check  the  result  by  comparison  with  the  statements  of  the 
problem. 


ADDITION  AND  SUBTRACTION  61 

EXERCISE  24 

1.  What  number  increased  by  11  e(|uals  19  ? 

2.  There  are  two  numbers  of  which  the  larger  is  5  times 
the  smaller.  The  difference  between  the  numbers  is  24.  Find 
the  numbers. 

3.  One  number  exceeds  another  by  54.  The  larger  number 
is  7  times  the  smaller.     Find  the  numbers. 

4.  Five  times  a  certain  number  exceeds  8  by  37.  Find  the 
.number. 

6.  If  12  times  a  certain  number  is  diminished  by  8,  the  result 
is  the  same  as  if  4  times  the  number  is  increased  by  5.  Find 
the  number. 

6.  15  exceeds  twice  a  certain  number  by  the  same  amount 
that  3  times  the  number  exceeds  10.     Find  the  number. 

7.  The  age  of  John  is  double  that  of  his  brother  James. 
What  are  their  ages  if  equal  results  are  obtained  by  subtracting 
5  years  from  John's  age,  and  adding  10  to  James'  age. 

8.  The  age  of  A  is  twice  that  of  B,  and  the  age  of  C  equals 
the  sum  of  the  ages  of  A  and  B.  The  sum  of  the  ages  of  A 
and  C  exceeds  the  age  of  B  by  40  years.     Find  their  ages. 

9.  One  angle  is  four  times  as  large  as  a  second  angle ;  if 
their  sum  is  increased  by  5°,  the  result  is  one  straight  angle. 
(See  §  13.)     Find  the  angles. 

10.  A  farmer  wishes  to  inclose  a  rectangular  field  for  a  past- 
ure, making  it  15  rods  wide.  He  wants  to  make  it  as  long  as 
possible,  using  172  rods  of  wire  fencing,  which  he  has  on  hand. 
How  long  can  he  make  it  ? 

11.  If  six  times  the  area  of  Lake  Superior,  the  largest  fresh- 
water lake,  be  decreased  by  12,000  square  miles,  the  result 
equals  the  area  of  the  Caspian  Sea,  the  largest  salt-water  lake, 
180,000  square  miles.     Find  the  area  of  Lake  Superior. 


62  ALGEBRA 

12.  If  twice  the  height  of  Mt.  McKinley,  the  highest  moun- 
tain in  North  America,  be  decreased  by  11,798  feet,  the  result 
equals  the  height  of  Mt.  Everest,  the  highest  mountain  in  Asia, 
29,002  feet.     Find  the  height  of  Mt.  McKinley. 

13.  The  longest  river  in  the  world  is  the  combined  Mis- 
sissippi-Missouri and  the  next  longest  is  the  Nile.  Twice  the 
length  of  theNile  diminished  by  2800  miles  equals  the  length 
of  the  Mississippi ;  the  length  of  the  Mississippi  is  700  miles 
more  than  that  of  the  Nile.     Find  the  length  of  each  river. 

14.  The  highest  velocity  of  wind  recorded  in  the  United 
States  up  to  January  1,  1910  exceeded  25  times  the  lowest 
average  velocity  at  any  point  in  the  United  States  by  2  miles 
per  hour  j  the  highest  velocity  exceeds  the  lowest  by  98  miles. 
Find  the  highest  and  lowest  velocity.     . 

15.  In  the  United  States,  the  lowest  average  annual  precip- 
itation, rain  and  snow,  is  at  Yuma,  Arizona ;  and  the  highest 
is  at  Mobile,  Alabama.  The  precipitation  in  inches  at  Mobile 
is  20  times  that  at  Yuma ;  the  sum  of  the  two  precipitations  is 
65.1  inches.     Find  the  precipitation  at  each  place. 

44.  Percentage  and  Interest  Problems.  Many  of  the  problems 
involving  percentage  and  interest  may  be  expressed  and  solved 
by  algebraic  methods. 

EXERCISE  25 

Percentage  Problems 

1.  What  does  4  %  mean  ?     5%?     r%?     m%? 

2.  What  is  4  %  of  500  ?     6  %  of  250  ? 

3.  Express  decimally  4  %  of  p ;  6  %  of  6 ;  15  %  of  c. 

4.  If  the  cost  of  an  article  is  c  dollars,  and  the  rate  of 
gain  is  25%,  what  is  the  gain?  what  is  the  selling  price? 
(c-f-.25c=?.) 

5.  Find  the  cost  of  an  article  sold  for  $  165  if  the  gain  is 
10%.     (c  +  . 10  c  =  165.) 


ADDITION   AND  SUBTRACTION  58 

6.  A  grocer  wishes  to  make  25  %  on  some  canned  goods. 
At  what  price  must  he  buy  them  so  as  to  be  able  to  sell  the 
goods  at  $1.25  per  dozen  ? 

7.  A  man  wishes  to  sell  hats  at  $3.50  each.  At  what  price 
must  he  buy  them  so  as  to  make  12  %  upon  the  cost? 

8.  A  real  estate  agent  knows  that  he  can  sell  a  certain  lot 
for  S3270.  At  what  price  must  he  buy  the  lot  from  the 
present  owner  in  order  that  he  may  make  a  profit  of  9  %  ? 

Interest  Problems 

9.  What  is  the  simple  interest  on  $200  at  6  %  for  1  year? 
for  2  years  ?  for  t  years  ? 

10.  What  is  the  simple  interest  on  R  dollars  at  6  %  for 
1  year  ?  for  2  years  ?  for  t  years  ? 

11.  What  is  the  simple  interest  on  $200  at  r  %  for  1  year  ? 
for  4  years  ?  for  t  years  ? 

12.  What  is  the  simple  interest  on  P  dollars  at  r  %  for 
1  year*?     for  3  years  ?     for  t  years  ? 

13.  If  /  represents  the  number  of  dollars  interest  on  P 
dollars  invested  at  r  %  for  t  years,  it  may  be  expressed  by  the 

Prt 

formula:      /  = .      Express  this  formula  in  words. 

100  ^ 

Solve  the  following  problems  by  substituting  in  this  formula : 

14.  If  a  man  receives  $  1150  income  from  $  3500  which  has 
been  invested  for  4  years  and  6  months,  what  rate  of  interest 
has  he  received  ? 


(Since  I  =  P .  -^  ■  t.) 

^  100       ^ 


Solution  : 

1.  Let         r  =  the  rate  per  cent. 

2. 

P  =  3500;  7=1150;  t  =  ^. 

3. 

35        , 
.-.1150  =  -3606-.    ^   .44.       (S 
100-     ^ 

.-.  1150  =  1  .  35  .  r. 

11.50  =q^r. 

4.M2: 

2300  =  315  r. 

6.  Dsis  : 

r  =  7.3+ 0/0. 

64  ALGEBRA 

15.  What  principal  must  be  invested  at  4  %  to  yield  an 
income  of  $  1500  per  year  ? 

16.  For  what  length  of  time  must  $4000  be  invested  at 
5  %  simple  interest  to  yield  $  750  interest  ? 

17.  What  rate  of  simple  interest  has  been  earned  on  an 
investment  of  $  2500,  if  the  income  is  $  1000  in  10  years  ? 

A7nou7it  in  Interest  Problems 

18.  The  sum  of  the  principal  and  interest  is  the  amount. 
Indicate  the  amount  at  the  end  of  one  year  if  P  dollars  are 

invested  at  4  %.     (P  +  .04  P  =  1.04  P.) 
Similarly  at  6  %  ;  at  7  %. 

19.  What  is  the  amount  at  the  end  of  Wo  years  if  P  dollars 
are  invested  at  4  %  ?  for  3  years  at  5  %  ? 

20.  What  sum  of  money  will  amount  to  $  3500  if  invested 
at  5  %  simple  interest  for  5  years  ? 

21.  How  long  will  it  take  $  1500  to  amount  to  $  2000  if 
invested  at  5  %  ?  (2000  =  1500  +  1500  •  yf  <y  .  t.  Sol,ve  the 
equation  for  t.) 

22.  How  long  will  it  take  $  1200  to  double  itself  at  6  %  ? 

23.  Letting  A  represent  the  number  of  dollars  in  the 
amount,  and  P,  r,  and  t  the  usual  numbers,  show  that  the 
amount  may  be  expressed  by  the  formula : 

Solve  the  following  problems  by  substituting  in  this 
formula : 

24.  In  how  many  years  will  $  3500  amount  to  $  4550  at  5  % 
simple  interest  ? 

25.  The  members  of  a  certain  company  paid  $  300  per 
share  for  some  stock.  At  the  end  of  7  years,  they  received 
$  800  per  share  for  their  stock.  What  rate  of  simple  interest 
did  their  money  earn  for  them  during  that  time  ? 

HisTOKiCAL  Note.  —  The  symbol  %  was  used  first  about  the  year  1685. 


IV.   PARENTHESES 

45.  Terms  which  are  parts  of  a  single  number  expression 
are  often  inclosed  in  Symbols  of  Grouping. 

The  Parentheses,  (  ),  are  symbols  of  grouping;  thus, 

3  a  —  {2x-\-y  —  z) 

means  that  2  x  -\-y  —  2;  is  to  be  subtracted  from  3  a. 

Other  symbols  of  grouping  are  the  Brackets  [  ],  the 
Braces,  j    j,  and  the  Vinculum, . 

All  are  used  in  the  same  manner  as  the  parentheses,  and  are 
referred  to  collectively  as  parentheses. 

Historical  Note. — The  parentheses  are  used  most  commonly  now. 
They  were  introduced  by  Girard,  a  Dutch  matheiuaticiau,  about  1G29. 
Previously  the  brackets  and  the  braces  had  been  used  by  Vieta,  about 
1593,  altliough  Bombelli,  an  Italian,  had  made  the  first  start  in  the  direc- 
tion of  their  use  in  1572.     Descartes  used  the  vinculum. 

REMOVAL  OF  PARENTHESES 

46.  Parentheses  preceded  by  a  Plus  Sign. 
Dkvelopmkxt.     1.    Consider  a^  +  a-  -f  (a^  —  a). 
This  means  that  a^  —  a  is  to  be  added  to  a^  +  a^. 

a^  -a  =  a*  -\-  a^  +  a^  ^  a. 

Sum :  a*  +  a^  +  cC^  —  a 

2.    Consider  a  —  h  -^{c  —  d). 
Solution  : 

a  —  b  ,-,  a  -h  +(c  —  d) 

4-  c  —  (7  =a  —  b  +  c  —  d. 

Sum :  a^b  +  c  —  d 

66 


56  ALGEBRA 

3.  Notice  that  the  signs  of  the  terms  which  are  within  the 
parentheses  are  not  changed  when  the  parentheses  are  re- 
moved.    This  fact  suggests  the 

Rule.  —  To  remove  parentheses  preceded  by  a  plus  sign : 
Rewrite  all  terms  which  are  within  the  parentheses  without 
changing  their  signs. 

47.   Parentheses  preceded  by  a  Minus  Sign. 
Development.     1.   Consider  a^  -\-a-  —  (a^  —  a). 
Solution  :  This  means  that  (a^  —  a)  is  to  be  subtracted  from  a*  +  a^. 
a*  +  a2  ...  ^4  _,.  ^2  _  (^_  ^3  _  a) 

+  a^  -a  =a*  -h  a^  —  (^  -{-a. 


Eemainder :  a^  —  a^  +  a^  +  a 

2.   Consider  a  —  &— (4-c—  d). 

Solution  : 

a-h 

-\-  c—  d 

.-.  a-5-(+c-d) 
=  a-h-c  +  d. 

Eeniainder :  a  —  b  —  c  +  d 

3.  Notice  that  the  signs  of  the  terms  which  are  within  the 
parentheses  are  changed  when  the  parentheses  are  removed. 
This  fact  suggests  the 

Rule.  —  To  remove  parentheses  preceded  by  a  minus  sign : 
Rewrite  all  terms  which  are  within  the  parentheses,  but  change 
their  signs  from  -h  to  - ,  or  from  -  to  + . 

Example  1.    Remove  the  parentheses  from 

2  a  -  S  b  -  (5  a  -  4  b)  +  (4.  a  -  b). 
Solution  :  By  the  rules,  the  expression  becomes 
2a- 36 -5a  +  4  6+4  a -6. 
When  this  is  simplified,  the  result  is  a. 

Check:  Let  a  =  1,  and  6  =  2. 

2  a  -  3  6  -  (5  a  -  4  6)  +  (4  a  -  6)  =  2  -  6  -  (5  -  8)  +  (4  -  2) 

=  2-6-(-3)  +  2 
=  2-6  +  3  +  2=+ 1, 
Also  the  result  a  =  1. 


PARENTHESES  67 

48.  The  above  rules  apply  equally  to  the  removal  of  all 
symbols  of  grouping  (§  45). 

It  should  be  noticed  in  each  case  that  the  sign  of  the  first 
term  within  the  symbol  of  grouping  is  not  the  sign  prefixed  to 
the  symbol  of  grouping ;  thus,  in  —  («  —  6)  the  term  a  is 
positive  within  the  parentheses. 

49.  Parentheses  sometimes  inclose  others ;  in  this  case,  the 
following  rule  should  be  observed  by  beginners. 

• 
Rule.  —  To  remove  two  or  more  parentheses,  when  one  incloses  an- 
other : 

1.  Combine  the  terms  within  the  innermost  parentheses,  if  pos- 
sible ;  then  remove  these  parentheses  according  to  the  rules  in  §§46 
and  47. 

2.  Combine  the  terms  within  the  resulting  innermost  parentheses, 
and  remove  these  parentheses. 

3.  Continue  doing  this  until  all  parentheses  are  removed. 

Example  2.     Simplify  4  a;—  J3x  +  (—  2  a;—  [a;  — a])j. 

The  brackets  are  the  innermost  symbols  of  grouping ;  they  are  pre- 
ceded by  a  minus  sign.  Remember  that  the  sign  of  x  within  the  brackets 
is  +. 

SOLDTION  : 

1.  4a;-{3a;-f(-2a;-[a;-a])} 

2.  =4a;-{3a;4-(— 2x  — x  +  a)}  Removmg  the  [  ]. 

3.  =  4ic— (3ic -f  (— 3  a;  +  a)}  Combining  within  the  (  ). 

4.  =4a;— {3  a;  — 3x  4- a}  Removing  the  (  ). 

5.  =  4  X  —  a.  Combining  within  the  {  }. 

EXERCISE  26 

Simplify  the  following  expressions  by  removing  the  symbols 
of  grouping : 

1.  3m-(2?i-|-p).  4.   7  v-h(-3«>  +  ic). 

2.  ^x-\-{z-2y).  6.   6r-[-3a-f&]. 

3.  5a-(26-c).  6.   8s  +  {-3m-i{. 


58  ALGEBRA 

7.  9k-[p-2q'].  9.    l-lm-n\. 

8.  2a-\-{3b-c).  10.   S-[-2a  +  b]. 

In  the  following  examples,  since  neither  symbol  of  grouping 
incloses  the  other,  remove  both  at  once.  Tell  what  each 
example  means;  thus  in  11:  "'subtract  the  number  (c  —  d) 
from  the  number  (2  a  —  h)P 

11.  {2a-h)-{c  —  d).  16.    {x-y)-{z- id). 

12.  (a-h)-{c  +  d).  ^        17.    [m  +  7.]  +  [i^-r7]. 

13.  [r  +  s]-fp-'y].  18.    \m-\-n\-\p -q\. 

14.  \r-s\  —  \t-\-v\.  19.    {-a  +  h)-{-{-c+d). 

15.  {x-y)-{z-\-io).  20.    {-a  +  h)-{-c-d). 

In  the  following  examples,  combine  terms  after  removing 
the  parentheses : 

21.  3  c- (5  c -6).  26.  (3  6- ll)-(4  6  + 5). 

22.  (2«-6)-(5a--6).  27.  {^x-\-y)-{-2x-]-y). 

23.  2a-(3a-6)  +  26.  28.  bx''-{-2x' +  x)  +  Q. 

24.  a2_^a-(a2-a).  29.  (2  r- s) -{3r-2  s). 

25.  (5a;-6)-(2a;-4).  30.  (3i9-2g) -f  (-3p-2g). 

31.  9m  — (4m  +  6n)4- (3m  — m). 

32.  8r  +  (5r-[2s  +  q). 

33.  Sx-{2y-\-\px-y']). 

34.  5a-(4a-J3a-lS). 

35.  66-J76-(9&  +  4)-7|. 

36.  {lt-r)-l^t-{10r-{-t)-Sr]. 

37.  3a;-(5a;-[7a;-h9a-4]-3a). 

38.  5a6-[(3a6-10)-(4a6  +  7)]. 

39.  7c-(5c  +  [12c-S6c  +  2j]). 

40.  m  — [(6  m  —  7  7i)  —  w]  —  [3  m  4-  4  71  —  (2  m  —  3  n)]. 


PARENTHESES  69 

INTRODUCTION  OF  PARENTHESES 

50.  Sometimes  it  is  necessary  to  introduce  parentheses  into 
an  expression. 

Development.  1.  What  is  the  rule  for  removing  parenthe- 
ses preceded  by  a  phis  sign  ? 

2.  Wliat,  then,  should  be  done  with  the  signs  of  terms  which 
are  placed  ivithin  parentheses  preceded  by  a  plus  sign  ? 

3.  Inclose  the  last  two  terms  of  a-\-h-\-  c  —  d  in  parentheses 
preceded  by  a  plus  sign. 

4.  What  is  the  rule  for  removing  parentheses  preceded  by  a 
minus  sign  ? 

5.  What,  then,  should  be  done  with  the  signs  of  terms  which 
are  placed  vithhr  parentheses  preceded  by  a  minus  sign  ? 

G.  Inclose  the  last  two  terms  of  a  +  6  -f  c  —  d  in  parentheses 
preceded  by  a  minus  sign. 

Rule.  —  1.  To  inclose  terms  in  parentheses  preceded  by  a  plus 
sign,  rewrite  the  terms  without  changing  their  signs. 

2.  To  inclose  terms  in  parentheses  preceded  by  a  minus  sign,  re- 
write the  terms,  changing  the  signs  from  +  to  - ,  and  from  -  to  4- . 

"     *t      ^    \ 
Example.     Inclose  the  last  three  terms  of  ^  +  s  — ?-f  vMn 

])arentheses  preceded  by  a  minus  sign. 

Solution:  1.    r-]-s— t-{-v  =  r  —  (— s-{-t  —  v).     (Rule  2). 

Check  :  If  the  parentheses  are  removed,  the  result  is  the  original 
expression. 

EXERCISE  27 

Inclose  the  last  three  terms  of  the  following  expressions  in 
parentheses  preceded  by  a  plus  sign : 

1.  m-^n  —  c  +  d.  6.    a-  +  6^  —  2  6c  -f  cl 

2.  a^h-r-  s.  7.    a'  -h'^-\-2hc-  c". 
S.   x-i-y  +  z-w.  8.    a^- 4  62 +  12  6 -9. 

4.  r  —  s'-i-t  +  x.  9.   a^  —  2  2/2  +  2/^  +  z\ 

5.  p—q  —  a  +  b.  10.    ii*  —  5  ?t'^  —8n^-\-0  n  +  7. 


60  ALGEBRA 

11-20.    Inclose  the  last  two  terms  of  the  foregoing  expres- 
sions in  parentheses  preceded  by  a  minus  sign. 

20-30.    Inclose  the  last  three  terms  in  parentheses  preceded 
by  a  minus  sign. 

EXERCISE  28 

Indicate  and  simplify,  where  possible,  the  following : 

1.  The  sum  of  a  and  5 ;  a  and  b. 

2.  The  sum  of  x  and  (a?  +  3) ;  x  and  (x  —  5). 

3.  The  difference  between  a  and  5 ;  a  and  b. 

4.  The  difference  between  (a  —  2)  and  3  a;  (a  +  7)  and  3  a. 

5.  The  sum  of  (3  a  -f-  5)  and  (2  a  —  6)  ;   also  their  differ- 
ence. 

6.  The  amount  by  which  15   is  greater  than  12;  greater 
than  a ;  greater  than  (a  -\-  2). 

7.  The  amount   by  which  5  x  is  greater  than  10;    greater 
than  2  x ;  greater  than  (x  —  4). 

8.  The  amount   by  which  20  exceeds  15;    exceeds  t;  ex- 
ceeds (^  —  3). 

9.  The  amount  by  which  Sp  exceeds  5;  exceeds  (p  -{-1); 
exceeds  (p  —  3). 

10.  The  amount  by  which  (2  a  +  3)  exceeds  (a  —  5). 

11.  The  smaller  part  of  15,  if  I  is  the  larger  part. 

12.  The  smaller  part  of  x,  if  7  is  the  larger  part. 

13.  The  larger  part  of  w,  if  3  is  the  smaller  part. 

14.  The  smaller  part  of  2  a,  if  (a  —  3)  is  the  larger  part. 

15.  The   larger  part  of  (3  c  —  2),  if  (c  +  1)  is  the  smaller 
part. 

16.  The  sum  of  two  numbers  is  50 ;  the  smaller  is  p.     Eepre- 
sent  the  larger  number. 


PARENTHESES  61 

17.  The  sum  of  two  numbers  is  37;  the  larger  is  I.     Repre- 
sent the  smaller. 

18.  The  difference  between  two  numbers  is  5 ;  the  smaller 
is  s.     Represent  the  larger. 

19.  The  difference  between  two  numbers  is  x-,  the  smaller 
is  3.     Represent  the  larger. 

20.  The  difference  between  two  numbers  is  10 ;  the  larger 
is  n.     Represent  the  smaller. 

21.  The  integer  which  is  consecutive  to  the  integer  a. 
Hint  :  6  is  consecutive  to  5  ;  7  is  consecutive  to  6. 

22.  The  second  integer  consecutive  to  a. 

23.  The  sum  of  a  and  the  consecutive  integer. 

24.  The  sum  of  n  and  the  two  consecutive  integers. 

25.  If  a  is  an  odd   number,  what  is   the   consecutive  odd 
number  ? 

Hint  :  5  is  consecutive  to  3  j  11  to  9  ;   17  to  15. 

51.   Parentheses  are  used  in  equations  and  problems. 

Example.     The  sum  of  two   numbers  is  88.      The  larger 
number  exceeds  the  smaller  by  36.     Find  the  numbers. 

Solution  :  1.     Let  I  =  the  larger  number. 


(§47) 


2.    Then 

(88  —  Z)  =  the  smaller  number. 

3.    Then 

i  _  (88  -  0  =  86. 

4.    Removing  (  ), 

Z  _  88  +  Z  =  36. 

2  i  -  88  =  36. 

5.    Agg:* 

2  I  =  124. 

6.    D2: 

I  =  62,  the  larger  number. 

88  -  62  =  26,  the  smaller  number. 

Check: 

62  -  26  =  36. 

*  See  §  42  for  the  symbol  Age- 


62  ALGEBRA 

EXERCISE  29 

Solve  the  following  equations 

1.  2r-(r4-6)=ll. 

2.  (3a;  +  4)-(2a;+9)  =  15. 

3.  12A-\-(2A-S)=(5A-(-7A-15). 

4.  (4i_5)._(2^4.5)  =  (2^-9)-(4«-7). 

5.  (-  10  a;  +  12)  -  (6  a^  -  5)  =  (13  -  8  x)  -  (15  +  9  x). 

6.  (3.T  +  5)-(2a;-7)  =  (4a;  +  9)-  (2a;-ll). 

7.  (6y/-8)-(9  2/+4)  =  (9-132/)-ll2/. 

8.  (13  -4m)-  (3  m  -h  9)  =  (2  7ii  -  8)  -  (6  +  7  m). 

9.  (8^-ll)-(7-50  =  12-(13  +  40. 

10.  (17  ;s  - 1)  -  (9 ^ - 10)  =  15  -(13  2  +  6)  -  22. 

11.  The  sum  of  two  numbers  is  30.  The  greater  number 
exceeds  the  smaller  number  by  4.     Find  the  numbers. 

12.  The  sum  of  the  ages  of  A  and  B  is  115  years.  A  is  13 
years  younger  than  B.     What  are  their  ages  ? 

13.  Divide  the  number  123  into  two  parts  such  that  the 
greater  exceeds  the  smaller  by  67. 

14.  The  sum  of  the  ages  of  A  and  B  is  102  years.  A  is  26 
years  older  than  B.     Find  their  ages. 

15.  Divide  $93  between  A  and  B  so  that  A  may  receive  $23 
less  than  B. 

16.  The  Library  of  Congress  at  Washington  consists  of  three 
main  stories,  whose  total  height  is  64  feet.  The  height  of  the 
second  story  exceeds  that  of  the  first  by  7  feet,  and  is  8  feet 
less  than  that  of  the  third  story.  Find  the  height  of  each  of 
the  three  stories. 

17.  The  total  length  of  the  Upper  Steel  Arch  Bridge  at 
Niagara  Falls  is  1240  feet.  Its  main  span,  the  longest  of  the 
kind  in  the  world,  exceeds  twice  the  remaining  part  of  the 
bridge  by  40  feet.     Find  the  length  of  its  main  span. 


parenthesp:s  63 

18.  The  first  appropriation  for  a  library  for  Congress  was 
made  in  1800.  The  present  Library  of  Congress  was  completed 
at  a  total  expense  of  over  $  6,000,000.  The  cost  of  the  gold 
leaf  on  the  dome  of  the  present  building  increased  by  $1200 
equals  the  appropriation  made  in  1800 ;  and  the  sum  of  that 
first  appropriation  and  the  cost  of  the  gold  leaf  is  $  8800.  Find 
each  of  these  amounts. 

19.  Seven  major  planets  besides  the  earth  revolve  around  the 
sun.  The  number  of  planets  which  are  farther  from  the  sun 
than  the  earth  exceeds  by  one  twice  the  number  which  are 
nearer  to  the  sun  than  the  earth.  Find  the  number  of  planets 
nearer  to  and  the  number  farther  from  the  sun  than  the  earth. 

20.  Tlie  sum  of  two  consecutive  integers  is  35.  Find  them. 
(See  Example  21,  Exercise  28.) 

21.  The  sum  of  three  consecutive  integers  is  108.  Find  the 
numbers. 

22.  The  sum  of  four  consecutive  integers  is  218.    Find  them. 

23.  The  sum  of  two  consecutive  odd  integers  is  196.  Find 
them. 

24.  Find  the  integer  which  is  such  that  when  increase.d  by 
the  first  consecutive  integer,  and  their  sum  decreased  by  the 
second  consecutive  integer,  the  result  is  75. 

25.  There  are  four  angles  which  make  up  the  total  angular 
magnitude  around  a  point.  (See  §  13,  Example  15).  The  second 
angle  is  3  times  the  first ;  the  second  exceeds  the  third  by  10°, 
and  exceeds  the  fourth  by  50°.  Find  the  angles.  Illustrate 
with  a  figure. 

26.  An  angle  exceeds  its  supplement  by  30°.  Find  the  angle. 
(See  Example  11,  §13). 

27.  The  complement  of  a  certain  angle  exceeds  the  angle 
itself  by  20°.     Find  the  angle.     (See  §  13.) 

28.  The  sum  of  the  supplement  and  complement  of  a  certain 
angle  is  120°.     Find  the  angle. 


64  ALGEBRA 

29.  The  fertility  of  farm  land  is  maintained  by  adding  fer- 
tilizers which  contain  certain  plant  foods,  such  as  nitrogen, 
potash,  and  phosphoric  acid.  In  100  pounds  of  a  good  corn 
fertilizer,  the  amount  of  phosphoric  acid  should  exceed  the 
amount  of  nitrogen  by  10  pounds,  and  the  amount  of  potash 
should  equal  in  weight  the  sum  of  the  other  two.  Find  the 
number  of  pounds  of  each  plant  food  in  the  mixture. 

30.  Texas  is  the  largest  state  in  the  Union,  and  Rhode 
Island  is  the  smallest.  The  area  of  Texas  exceeds  213  times 
the  area  of  Khode  Island  by  72  square  miles.  The  sum  of 
their  areas  is  267,144  square  miles.  Find  the  area  of  each 
state. 

Historical  Notk.  —  In  Examples  5  and  8,  negative  roots  are  found. 
The  mathematicians  of  the  16th  century  were  slow  to  admit  that  such 
roots  had  any  meaning.  Cardan  called  them  numerce  fictce.  Even  such 
men  as  Vieta  and  Harriot,  who  contributed  so  much  to  the  growth  of 
algebra,  admitted  only  positive  solutions.  Girard  (1590-1632)  and  Des- 
cartes (1673)  were  especially  instrumental  in  establishing  the  fact  that 
negative  roots  should  have  the  same  meaning  as  positive  roots. 


V.   MULTIPLICATION 

52.  The  Law  of  Signs  for  Multiplication  of  positive  and  nega- 
tive numbers  (§  25),  may  be  written: 

Rule.  —  1.  The  product  of  two  numbers  having  like  signs  is  posi- 
tive. 

2.   The  product  of  two  numbers  having  unlike  signs  is  negative. 

Historical  Note.  —  Until  the  time  of  Michael  Stifel  (looS),  little 
progress  was  made  toward  finding  the  rule  for  multiplying  powers  of  the 
same  base,  owing  to  the  cumbersome  notation  that  had  been  used  pre- 
viously to  denote  the  various  powers.  Some  men  even  made  a  table  cor- 
responding to  our  multiplication  table,  giving  the  products  of  some  of  the 
powers.  Stifel  introduced  a  better  notation,  used  the  word  exponent, 
and  gave  the  rule  which  we  now  use. 

53.  The  Law  of  Exponents  for  Multiplication. 
Development.     1.    Review    the    definitions    of    exponent, 

base,  and  power  of  a  number,  giv^en  in  paragraph  15. 

2.  What  does  a^  mean?  ^^'^  r^?  e^? 

3.  Write  in  exponent  form  : 

(a)  b'b'b'b;    (b)  m-m-m-,    (c)  y  •  y  -  y  •  y  -  y; 

(d)  r  •  r  .  • .  r  (if  there  are  10  factors) ; 

(e)  a;  •  a;  •  •  •  a;  (if  there  are  8  factors). 

4.  Find  the  product  of  a^  and  a*. 

a^  =  a-  a  •  a;  a*  =  a  ■  a  -  a  •  a. 
.'.  a^  •  a*  =  (a  •  a '  a)  X  (a  •  a  -  a  •  a)  =:  a   a  •  a  '  a  •  a  '  a  '  a  =  d^ . 

5.  Find  as  in  4,  the  following  products  and  write  down  the 
results  as  in  part  (a) : 

(a)  a^'a*  =  a!.  (c)  r"  -1'''  =  '^       ' 

(6)  a^  .  if*  =  ?  ((T)  m^  •  m-'  =  ? 

66 


66  ALGEBRA 

6.  Observing  the  results  in  5,  see  if  you  can  get  the  follow- 
ing products  mentally;  check  as  in  4: 

(a)  m^  '  m^  =  ?  (c)  x^  -  a^  =  ? 

(b)  m^ .  m^  =  ?  (d)  f  •  2/'  =  ? 
,  7.    The  facts  observed  lead  to  the 

Rule.  —  To  find  the  exponent  of  any  number  in  a  product,  add  the 
exponents  of  that  number  in  the  multiplicand  and  multiplier. 

EXERCISE  30 

Find  the  indicated  products 

1.  X- .  x\         4.  t"  '  t\  1.  m^^  •  m\  10.  2^  •  2\ 

2.  a'  .  a\         5.  f  •  t\  8.  s^'  •  s\  11.  3^  •  3^ 

3.  iiv"  .  ml        6.  f  '  t\  9.  if  .  y\  12.  5^ .  51 

54.   The  Commutative  Law  of  Multiplication. 
Development.     1.    3x4  and  4x3  each  equal  12. 

2.  Notice,  in  step  1,  that  the  same  factors^  occur  in  each 
product,  but  that  the  order  is  changed.  Does  changing  the 
order  of  the  factors  change  the  value  of  the  product  ? 

3.  Comjjare  3x4x5  and  3  x  5  x  4,  by  finding  their  values. 

4.  Write  these  same  factors  in  some  other  order,  and  com- 
pare the  product  with  the  products  obtained  in  step  3. 

5.  Arrange  the  factors  2,  4,  and  5  in  three  different  ways 
and  compare  the  products. 

Rule.— The  factors  of  a  product  may  be  arranged  in  any  order 
without  changing  the  value  of  the  product. 

EXERCISE  31 

Arrange  each  of  the  following  sets  of  factors  in  two  ways 
and  find  the  products. 

1.  3,  7.  4.   5,  2,  4. 

2.  2,  5,  6.  5.    6,  5,  3. 

3.  3,  4,  6.  6.    6,  4,  2. 


MULTIPLICATION  67 


MULTIPLICATION  OF  MONOMIALS 

55.   Development.     1.   Find  the  product  of  7  a  and  —2  b. 
Solution:!.  -  2  6  =  (- 2)6.  (See  §31) 

2.  Then  7  «(- 2  6)=  7  •  a  •  (- 2)  .  6 

Then,  since  the  order  of  the  factors  may  be  changed,  (§  64) 

3.  7a(-2  6)  =  7.  (-2).rt.6 

=  -  14  ab.  (§  6) 

2.  Find  the  product  of  —ox  and  —  6  x^y. 
Solution:  1.    (- 5x)(- 6x^y)=  (- 6)  ■  x  -  (- 6)  .  x'^ -y. 

=  (-">)-<-«)-^-^'^-y 

=  +  30  x% 

3.  Find  similarly  the  product  of  3  mn  and  2j\ 

4.  Find  similarly  the  product  of  +  2  a^b  and  +  5  ab. 
6.   Give  at  sight,  if  possible,  the  following : 

(a)  2  r.s  *5t.  (c)  7  a  •  3  a". 

(b)  3  xy  '  2z.  (d)  4  m  •  5  mx. 

Rule.  —  To  find  the  product  of  two  monomials : 

1.  Find  the  product  of  the  numerical  coefficients,  using  the  Law 
of  Signs  for  Multiplication. 

2.  Multiply  this  product  by  the  literal  factors,  giving  to  each  an 
exponent  equal  to  its  exponent  in  the  multiplicand  plus  its  exponent 
in  the  multiplier. 

Example  1.     Multiply  2  a'  by  9  a*b\ 

Solution  :  (2  a5)(9  a*b^)  =  +  (2x9).  a^^+*)  ■  b^ 

=  18  a«&2. 

^     Example  2.     Multiply  —  5  x^yz!^  by  -f  6  xy^w. 

Solution  :     (  -  5  ofiijz^)  (  +  6  xy^w)  =  -(5x6).  x(8+i)  •  y(i+2)  .  z^  .  w 

=  -  30  x^y^z^w. 


^8  ALGEBRA 

EXERCISE  32 

Find  the  following  indicated  products: 

1.  af  •  x^  H.   a^yz  .  xy^z. 

2.  a^ .  a^ '  a\  12.    i^sH  .  rsE 

3.  x^  -oy^  •  x^  13.    a6V  .  aftV. 

4.  y^-y^-y^.  14.    rh  •  rs'e, 

5.  wi^ .  771  •  m^.  15.   2^  •  2^. 

6.  a}-  •  a^ .  a^  16.   4^  •  4^. 

7.  E^'E^'E,  17.    S^'.S*. 

8.  a'b^-ab.  ^  18.   10^ .  101 

9.  it-y  .  xy\  19.  2>r2^  •  r^. 
10.    a^x  •  aa^.                                   20.   mx^  •  mna^. 

Multiply :     ' 

21.  7  a"^  by  3  a^.  31.  -  6  a^b^  by  -  6V. 

22.  —  9  m^  by  8  m\  32.  +  9  ao^  by  —  9  a/. 

23.  —  5  a6  by  2  al  33.  8  a;V  by  -  8  2/V. 

24.  -  3  rs2  by  -  9  vs.  34.  -  7  AB^  by  +  12  ^5^(7. 

25.  5  xyz  by  —  11  aj^/s;.  35.  12  a-6c  by  6  bcd^ 

26.  - 11  a;/ by  9  a;^?/.  36.  13  a^  by  -  7  a^ft^^. 

27.  -  6  d'b  by  -  4  ab\  37.  -  14  m'n  by  -  6  mn^. 

28.  -  6  aj2?/3  by  +  12  xy\  38.  -  16  a^bc*  by  4-  5  a^6«c. 

29.  -  9  mhi^  by  7  mVl  39.  -  3  a^f^  by  -  17  x^fz. 

30.  - 12  rst  by  -  9  s2.  40.-4  m/i  by  + 17  m^n. 

Find  the  product  of : 

41.  3  a^,  5  a^,  and  6  a^. 

42.  -4.x^,  -9y%  and2;s^ 

43.  a^y^,  —  yh,  and  xz^. 

44.  - 12  a'b'',  -  b'c^,  and  -  8  cW 

45.  a^j  —3  a,  —  5  a%  and  —  4  a^. 


MULTIPLICATION 


46.    —  7  m^w,  +  8  mi-^,  and  —  2  nh\ 


47.    -3  and  -f-^^" 

i5 


48.    +  5  and 


'Zm-" 


49. 

4-  4  and  - 

3n 
2 

50. 

-f  12  and 

5r 
3  ' 

51. 

- 15  and 

Os 
5  ' 

52. 

- 10  and 

_  11 

63.    —  9  and  -j- 


4  J) 


54.    +14  and  -— . 

7 


55. 


18  and 


9  ' 


56.  +  20  and  -  j\  x. 

57.  ^-ai^and  -^a'b. 
58  —  f  mn  and  —  |  m^. 

59.  +  .5  rs^  and  -  .3  j-^s. 

60.  -  .12  f  and  -  .7  ^. 


56.   Numbers,  and  relations  between  them,  may 
be  represented  by  geometrical  figures. 

Example  1.     The  product  4x5  may  be  repre- 
sented by  the  rectangle  in  Fig.  1.     Note  that  the  ^ 
area  of  the  rectangle  is  20  square  units. 

Example  2.     The  product  4  (3  +  7)  may  be  represented  by 
the  Fig.  2. 


=  4 + 


Fio.  2. 

4(3  +  7)=4x3+4x7. 
4x10=    .12  +  28. 
Example  3.     The  product  a{b-{-c)  may  be  represented  by 
the  Fig.  3. 


ab 


ab 


b    +• 


Fi<;.  3. 
a  (6  -f  c)  =  a6  -h  rtc. 


70  ALGEBRA 

EXERCISE  33 

1.  Draw  a  figure  representing  5(3  +  4),  as  in  Example  2. 

2.  Draw  a  figure  representing  4(2  +  3  +  5),  as  in  Example  2. 

3.  Draw  a  figure  representing  a(6  +  c  +  cZ),  as  in  Example  3. 

MULTIPLICATION  OF   POLYNOMIALS  BY  MONOMIALS 

57.    Development.     1.    From  the   geometrical   illustration 
in  §  56,  it  is  clear  that :  ^ 

(a)  5(7  +  3)=(5x7)  +  (5x3)  =  35  +  15  =  50. 
(P)  6(4  +  5)  =  (6x4)+(6x5)  =  24  +  30  =  54. 
(c)  a(b-{-c-\-  d)  =  ah  -\-  ac-\-  ad. 

In  every  case,  each  term  of  the  polynomial  is  multiplied  by 
the  monomial. 

Rule.  —  To  multiply  a  polynomial  by  a  monomial : 

1.  Multiply  each  term  of  the  polynomial  by  the  monomial. 

2.  Unite  the  results  with  the  proper  signs. 

Example.   Multiply  3  a' -2  ab -\-b' hy  -3  ah. 

Solution  :  (3  a2  _  2  ah  +  6-^)  x  (-  3  «6)  =  -  9  a^ft  +  g  a%'^  -  3  ah^. 

Check  :  This  result  is  true  for  any  values  of  a  and  h.     Let  a  =  1,  and 
&  =  1. 

3a2_2rt&  +  /;2^3_2  +  i  =2,  -3a6=-3,  and  2  •  (- 3)  =  -6  ; 
also,  _9a36+6a26-^-3a&3  =  _94.6_3=-12  +  6  =  ~6. 

EXERCISE  34 
Multiply : 

1.  4  a  —  9  by  5  a.  6.  r-  —  2  7's  +  s^  by  —  i^s^. 

2.  m^  —  mn  +  v?  by  mn.  7.  G  .t^  —  5  x^  —  7  ic^  b}^  —  7  a?. 

3.  3ar^  +  iB-5  by  -9a;-.  8.  -  3  c^  -  d^ -\- 6  cd  hy  4.  c\l\ 

4.  8  0^2/  —  5  ^if  by  —  3  xif.  9.  —  3x-y-\-i(?—3  xy-  by  —x-y. 

5.  2a3~6a2-7by-7al  10.  a^ -;x^  ^xhy  -o?. 


MULTIPLICATION  71 

11.  5  m^  —  6mn  —  4  n^  13.    3c^-5xy-{-y^ 
3m^ -2ay 

12.  6p--5pq  +  9q^                14.    a^  -  3  a^b -{- S  ab^  -  b^ 
4:2Jq —ab 

Perform  the  following  indicated  multiplications: 

15.  7x'(x-5).  19.    -3ab'(a'-2ab  +  b'). 

16.  -6ab'(10a^-7b').        20.    S  x"  -  (6a^-5  x-12). 

17.  x^'(a^-Ax^y''-\-i/).        21.    -^a^b^ -  (3a^-2ab-4:b^. 

18.  —  rs  '  (r^  —rs  +  .s^).  22.    —  5  ?n*  •  (8  m*  —  m^  —  3). 

23.  Simplify  the  expression  :  3(2  x  —  6)—  2(x  +  6). 

This  means  that  (2  x  —  5)  is  to  be  multiplied  by  3  ;  that  (a;  +  6)  is  to  be 
multiplied  by  2  ;  and  that  the  second  result  is  to  be  subtracted  from  the  first. 
Solution:        3(2  a;  -  5)- 2(a;  +  0)  =  (G  x  -  15)-(2x+12). 

=  6x-15-2x- 12. 

=  4  a; -27. 

In  the  following  examples,  firsl^  tell  what  each  means,  as  in 
Example  23;  then,  simplify. 

24.  5(6a  +  36)+4(5a-2  6).   27.   3(2  m  +  8)-2(6 -5  m). 

25.  2a{3x-y)-3a(2x-\-y).   28.   3  c(2  m  -  4)-6c(2  m +  4). 

26.  x(x  —  y)—y(x-^y).  29.    r()*^  —  s)-f-s(r  —  s^). 

Multiply  and  then  simplify  the  following: 

3,    ,2(^_2^  +  ^).  3.    15(^-^1). 

34.  i(2a^-Ax-\-6).  40.    -ix2(3a;^-24a:2^15)^ 

35.  6(^x2-^a:2/  +  i2/0-  ^^-   imw(i  m^--^  m?i+i  «^. 


72  ALGEBRA 


MULTIPLICATION  OF  A  POLYNOMIAL  BY  A  POLYNOMIAL 

58.   A  number  may  be  multiplied  by  (2  +  3)  by  multiplying 
first  by  2  and  then  by  3,  and  adding  the  products.     Thus, 
(2  +  3)  X  6  =  (2  X  6)  +  (3  X  6)  =  12  +  18  =  30,  for  (5)  x  6  =  30. 
Similarly,  (2  +  3)  •  (a  +  &)  =  2(a  +  &)  +  3(a  +  b) 

=  2a  +  26  +  3a  +  36  =  5a+5  6, 
for  (5)  .  (a  +  &)  =  5  a  +  6  6. 

The  multiplier  in  each  case  consists  of  the  sum  of  2  and  3 ; 
the  multiplicand  is  multiplied  separately  by  2  and  by  3,  and 
the  products  are  added.     This  illustrates  the 

Rule.  —  To  multiply  one  polynomial  by  another : 

1.  Multiply  the  multiplicand  by  each  term  of  the  multiplier. 

2.  Add  the  partial  products. 

Example  1.     Multiply  3a  —  46  by  2a  —  5  5. 
Solution  :  In  accordance  with  the  rule,  multiply  3a— 46  by  2a  and 
then  by  —  5  6,  and  add  the  partial  products.     A  convenient  arrangement 
is  suggested  by  the  arrangement  of  multiplication  problems  in  arithmetic. 
43  3a_    4  6 

12  2a-    56 

86  =    2  x  43  6a^-    Sab  =2a(Sa-ib) 

430  =  10x43  -  15  gb  4-  20  62  =  -  5  6(3  g  -  4  6) 

516  =  sum  6  g2  _  23  g6  +  20  6^  =  sum 

Note  :  In  arithmetic,  the  multiplication  proceeds  from  right  to  left  ;  in 
algebra,  the  multiplication  proceeds  from  left  to  right. 

Example  2.     Multiply  a^  —  8  a;^  —  2  a^a;  by  2  ic  +  a. 

Solution  :  It  is  convenient  to  arrange  the  multiplicand  and  multiplier 
in  the  same  order  of  powers  of  some  letter  (§  37)  and  to  write  the  partial 
products  in  the  same  order.  Leave  spaces  for  any  powers  which  may 
not  be  present  in  the  multiplicand. 

Arranging  the  expressions  according  to  the  descending  powers  of  g,  we 

^*^®'  g3  -  2  a'^x  -  8  a:8 

a  +2x 

a*  -  2  a%  -  8  ax» 

+  2  g3a;  -  4  a%2 -  16x* 

a*  -  4  g2a;2  _  g  gx^  -  16  x* 


MULTIPLICATION  73 

Check  :  This  result  should  be  true  for  all  values  of  a  and  x. 

Let  a  =  1  and  x  =  1. 

Then,  a8-2a2x-8x3  =  l-2-8=-9, 

a  +  2x=l-l-2  =  +  3, 
and  (_9)x(+3)  =  -27; 

also,        a*-4a2x2-8ax3-16x*  =  l-4-8-16=-27. 

EXERCISE   35 

Multiply : 

1.  a;4-3byaj  +  5.  11.  2  a  +  &  by  a -f  6. 

2.  r-7byr-4.  12.  Sc -2dhj  2  c +d. 

3.  2s -5  by  8-3.  13.  5  r  +  6  s  by  3r  -  2s. 

4.  3m  +  2bym-4.  14.  5x-2yhy  Sx- 4.y. 

5.  4«-9by<  +  3.  15.  6m-3i:>by4m  +  5p. 

6.  3x-f  7by  2a;-f3.  16.  7  i/ -  9  z  by  6  ^  +  Sz. 

7.  2m-|-5by  5m-l.  17.  11  a+ 5  d  by  6  a- 4  d. 

8.  6i)-3by2i9  +  7.  18.  12p  +  7  g  by  8p  -  7  ^. 

9.  5y-lhy(jy-S.  19.  2x' -fhy  x'' -Sy\ 

10.    7z4-10by42-5.  20.   9  lo' -7  v  hj  11  w' +  Sv. 

21.  m^  — m  — 3bym  +  3. 

22.  2  a^  -h  7  a  —  9  by  5  a  —  1. 

23.  x'-2xy+Sy^hyx-Sy. 

24.  ar^-a;?/  +  2/2by  X4-2/- 

25.  «'-  +  4  a;?/  -h  16  2/2  by  a;  -  4  y. 

26.  m^  +  mn  4-  n^  by  m^  —  mn  +  n\ 

27.  rt3^^_2-2a2by  a2  +  2a-3. 

28.  3  -I-  a-'  -  7  a  -  4  a^  by  2  a  4- 1. 

29.  9a;4-2a^-5by44-3«'^-7a:. 

30.  6n-8  +  4n2by  -4  +  2n2-3n. 

31.  9r^-5y'  +  eryhySry+4:y'-^7r'. 

32.  3a'-5ab-Sb'hy4:a'-9ab-7b', 


41. 

ia-i6byia-f-.i&. 

42. 

^m  —  ^7ihj  ^m  —  ^ n. 

43. 

\x-\yhj  \x  +  ^y. 

44. 

2a-\hhj  a+\h. 

45. 

lx-\yhjix-^y. 

74  ALGEBRA 

33.  a  —  h  +  chja—h—  Co 

34.  r  +  s  -|-  ^  by  r  —  s  —  ^. 

35.  2  n^  +  m^  +  3  mn  by  2  7i2  _  3  m^  4.  rn\ 

36.  a^  +  3  ah"-  -  3  a'b  -  b^  by  a'  +  '^^  -  2  a6. 

37.  m*-3m'^+9m2-27m4-81  by  m  +  a 

38.  Aa-\-6b+10chy2a-3b-\-5c. 

39.  ic^  +  4  a?^  +  8  aj  +  2  ar"'  +  16  by  x  —  2. 

40.  a-  +  6^  +  c^  +.a6  —  6c  +  ac  by  a  —  6  —  c. 

46.  (3x-5y. 

47.  (2m-3w)3. 

48.  (4.r  +  5)l 

49.  (2  a; -3)*. 

50.  (3  a -4.  by. 

Find  the  product  of  the  following: 

51.  a  -\-  3,  a  —  4,  and  a  +  2. 

52.  m  +  4,  2  m  —  3,  and  m  —  5. 

53.  a;  +  ?/,  flj-  —  a;?/  +  .!/^  ^^^  x^  —  y^. 

54.  m  +  71,  ?>i^  +  71^,  and  m  —  n. 

55.  a;^  -f  x?/  +  2/^  ^  —  xy  -\-  jf,  and  a^  —  3/^ 

PARENTHESES  IN  MULTIPLICATION 
59.   Example.     Simplify  (a  -  2  a:)=^  —  2  (3  a  +  a;)  (a  —  x). 

Solution:  To  simplify  this  expression,  first  multiply  {a  —  2x)  by 
itself  (§  15)  ;  second,  find  the  product  of  2,  3a  +  x  and  a  —  X]  third, 
subtract  the  second  result  from  the  first. 

1.    a-2ic  2.   3a  +  « 

g  —  2a;  a  —  x 

a^-2ax 

-  2  ax  +  4  x^ 
a2  —  4  ax  +  4  x2 

6  a2  -  4  ax  -  2  x2 


3a2+    ax 

-3ax- 

X2 

3  a2  -  2  ax  - 

X2 

2 

MULTIPLICATION  75 

3.  Then  (a -2a;)2-2(3a  +  a;)(a -x) 

4.  ={a^-Aax-\-ix^)-(Qa^-4ax^2x^) 

5.  =rt2_4aa;4.4a:2-6a2 +  4ax  +  2x2  (§47) 
0.                    =-5a2  +  6a;2.  Answer. 

Note.    Be  careful  to  place  the  results  of  steps  1  and  2  in  parentheses  as 
iu  step  4. 

EXERCISE  36 

Tell,  as  in  the  above  solution,  what  must  be  done  to  simplify 
the  following ;  then  simplify  : 

1.  (3a-j-8)(a-6)  +  (2a-h7)(4a-9). 

2.  (2m  H-  7)  (3 m  -  5)- (2  m  -  5) (3 m  +  7). 

3.  (a-2x){a-{-3x)-^(a-\-2x)(a-Sx). 

4.  (2a-36)2-4(a-6)(a  +  56). 

5.  2(/iH-3)(/i-2)-(/i  +  5)(/i-6). 

6.  5(a;-4)(a;  +  l)-3(a;-3)(x-h2). 

7.  2(3a;  +  2)(4a;-3)-(3a;-2)(4a;  +  3). 

8.  3(3a4-5)(2a-8)-2(4a-7)(a  +  G). 

9.  4(3«-2)(a;  +  6)-5(2a;-7)(a;  +  2). 
10.  (a-\-b)(a''-\-b')-(a-b)(a'-b'). 

USE  OF  MULTIPLICATION  IN  EQUATIONS 

60.    Example  1.     Seven  times  the  complement  of  a  certain 
angle  exceeds  twice  its  supplement  by  20°.     Find  the  angle. 

SoLUTioy  :     1.  Let  a  =  the  number  of  degrees  in  the  angle. 

2.  Then,  90  —  a  =  the  number  of  degrees  in  the  complement, 

3.  and,  180  —  a  =  the  number  of  degrees  in  the  supplement. 

4.  Hence,  7(90  -  a)  =  2(180  -  a)  +  20. 
6.   Multiplying,  630  -  7  a  =  360  -  2  a  +  20. 

6.  Combining,  630  -  7  a  =  380  -  2  a. 

7.  Seao  :  -  7  a  =  -  250  -  2  a. 

8.  A2«:  -6a  =-250. 

9.  M_i  :  5  a  =  250. 
10.  Dg:  a  =  50. 

Check  :  The  angle  is  one  of  50*^.     The  complement  contains  40°  and 
the  supplement,  130°.     Does  7  x  40°  =  2  x  130°  +  20°  ?    Yes. 


76  algp:bra 

Example  2.     Solve  the  equation, 

(2a  +  5)(3a-7)-(2a-5)(3a  +  7)=:4. 
Solution:  1.  (2a  +  5)(3a  -  7)-(2a  -  5)(3a  +  7)  =  4. 

2.  Multiplying,  (Ga^  +  a  -  35)-(6a2  -  a  -  35)  =  4. 

3.  Kemoving  (  ),  Ga^  +  a  -  35  -  6«2  _(.  «  ^.  35  =  4. 

4.  Combining  terms,  2  a  =  4. 

5.  D2:  a  =  2. 
Check:  Does  (2  •  2  +  5)(3  •  2  -  7)-(2 -2  -  5)(3  •  2  +  7)=  4? 
Does                                                (9).(-l)-(-l)(+13)  =  4? 
Does                                                                     _9-(-13)  =  4? 
Does                                                                            -  9  +  13  =  4  ?    Yes. 

Note  :  lu  Exercises  like  Example  2,  be  careful  to  put  the  products  obtaiued 
in  step  1,  in  parentheses  as  in  step  2. 

EXERCISE  37 

Solve  the  following  equations : 

1.  2(7^-3)  =  20.  2.   4(2/-5)-h7  =  15. 

3.  3(2a^-4)  +  2(aj-5)  =  5(x  +  l). 

4.  6(2-3a;)-h3=3(4x-5). 

5.  12-5(3a-2)  =  2(a-6). 

6.  2(v  +  9)  +  3('y-4)  =  16. 

7.  4(^-3)  +  3(2^-f 5)=33(4-i). 

8.  10-5(3Z-4)  =  6(3-2Z). 

9.  7^-6(2^-5)=6(6-^). 

10.  37i-2(2w-7)  =  3(n-2). 

11.  3(4m-5)-4(m-6)  =  3(m  +  17)-7. 

12.  (x-5)(x-\-6)-(x-\-S)(x-^)=0. 

13.  (2/-7)(2/  +  2)-(2/-9)(^  +  3)  =  0. 

14.  (2r  4-  3)  (3r -  5)-  6(r-  4)  (r-  3)  +  5  =  0. 

15.  3(2s-4)(s  +  7)-2(3s-2)(s  +  5)  =  -3(s-2). 


MULTIPLICATION  77 

16.  The"  sum  of  two  numbers  is  75.  The  larger  exceeds  the 
smaller  by  11.     Find  the  numbers. 

17.  The  sum  of  two  numbers  is  100.  If  four  times  the 
greater  be  diminished  by  22,  the  result  is  5  times  the  smaller. 
Find  the  two  numbers. 

18.  The  distance  from  New  York  to  Paris  exceeds  the  dis- 
tance from  New  York  to  London  by  280  miles.  Four  times 
the  distance  to  London  exceeds  three  times  the  distance  to 
Paris  by  2900  miles.     Find  the  two  distances. 

19.  One  number  exceeds  another  number  by  7.  If  6  times 
the  smaller  is  diminished  by  5  times  the  larger,  the  remainder- 
is  5.     Find  the  numbers. 

20.  Separate  60  into  two  parts  such  that  4  times  the  smaller 
shall  exceed  2  times  the  larger  by  30. 

21.  The  sum  of  two  numbers  is  80.  If  twice  the  greater  be 
decreased  by  12,  the  result  exceeds  4  times  the  smaller  by  4. 
Find  the  numbers. 

22.  The  Library  of  Congress  stands  upon  a  rectangular  base 
whose  perimeter  is  1620  feet.  The  length  exceeds  the  width 
by  130  feet.     Find  the  dimensions  of  the  building. 

23.  There  are  two  consecutive  numbers  such  that  the  sum 
of  twice  the  smaller  and  three  times  the  larger  is  78.  What 
are  the  numbers  ? 

24.  There  are  two  consecutive  integers  whose  product  ex- 
ceeds the  square  of  the  larger  by  20.     What  are  they  ? 

25.  Th&  total  population  of  Chicago,  Philadelphia  and 
Greater  New  York  (1910  Census),  was  8,501,174.  The  popu- 
lation of  Chicago  exceeded  the  population  of  Philadelphia  by 
636,275 ;  the  population  of  New  York  exceeded  twice  the  pop- 
ulation of  Chicago  by  396,317.  Find  the  population  of  each  of 
the  cities. 


78  ALGEBRA 

EXERCISE   38 
Algebraic  Expression 

1.  A  is  now  15  years  of  age.     Express  his  age : 

(a)  5  years  ago  ;  (6)  m  years  ago;        (c)  y  years  ago; 

(d)  8  years  from  now ;    (e)  m  years  from  now. 

2.  B  is  6  years  of  age.     Express  his  age : 

(a)  4  years  from  now  ;     (6)  ?7i  years  from  now ; 

(c)  6  years  ago ;  (d)  x  years  ago ;        (e)  ^  years  ago. 

3.  A  is  now  x  years  of  age.     B's  present  age  exceeds  the  age 
of  A  by  5  years. 

(a)  Express  B's  present  age ;     (6)  the  sum  of  their  ages. 

(c)  Express  the  age  of  each  10  years  ago. 

(d)  Express  the  age  of  each  10  years  from  now. 

4.  A  is  now  a  years  of  age;  B  is  twice  as  old. 
(a)  Express  B's  present  age. 

(h)  Express  the  age  of  each  3  years  ago. 

(c)  Express  the  age  of  each  7  years  from  now. 

(d)  Express  the  fact  that  B's  age  5  years  ago  was  3  times 
A's  age  at  that  time. 

5.  Express  the  value  of : 

(a)  (15  —  x)  pounds  of  tea  at  40  ^  per  pound ; 
(6)  X  pounds  of  tea  at  60  ^  per  pound ; 

(c)  the  entire  amount  of  tea. 

6.  Express  the  value  in  cents  of : 

(a)  X  nickels ;  (h)  2x  dimes  ;  (c)  3  a;  dollars ; 

(d)  4  x  quarters  ;  (e)  all  of  the  coins. 

7.  Express  the  value  in  cents  of : 

(a)  d  dimes;  (h)  (15  —  d)  quarters ; 

(c)  (15  4-  3  c?)  half  dollars  ;  (c^)  all  of  the  coins. 


MULTIPLICATION  •  79 

8.  Express  in  inches : 

(a)  m  feet ;         (6)  3  m  yards. 

(c)  the  combined  length  of  m  feet  and  3  m  yards. 

9.  Express  in  pints : 

(a)  3  X  pints  plus  2x  quarts  plus  5  x  gallons. 

(6)  2  c  pints  plus  (3  c  —  2)  quarts  plus  (5  —  3  c)  gallons. 

EXERCISE   39 

1.  The  sum  of  the  ages  of  A  and  B  is  50  years ;  in  5  years 
A  will  be  5  times  as  old  as  B.     Find  their  ages. 

Solution  :  1.     Let  a  =  the  number  of  years  in  A's  age  now. 

2.  Then        (50  —  a)  =  the  number  of  years  in  B's  age  noio. 

3.  Then  a  4-  5  =  the  number  of  years  in  A's  age  in  5  yr. 

4.  and  (50  —  a  +  5)  or  55  —  « 

=  the  number  of  years  in  B's  age  in  5  yr. 

5.  .-.  (a  +  5)  =5(65-rt). 
Complete  the  solution. 

Note.    Represent  with  care  the  present  ages  of  both  persons ;  also  their 
ages  at  the  other  time  mentioned ;  then  form  the  equation. 

2.  A  father  is  now  9  times  as  old  as  his  son.  In  9  years 
he  will  be  only  3  times  as  old  as  his  son.  What  are  their 
present  ages  ? 

3.  The  difference  between  the  present  ages  of  a  father  and 
son  is  25  years.  In  10  years  the  father  will  be  twice  as  old 
as  his  son.   .What  are  their  present  ages  ? 

4.  A  is  5  times  as  old  as  B.  In  9  years  he  will  be  only 
3  times  as  old  as  B.     What  are  their  ages  ? 

5.  B  is  twice  as  old  as  A.  35  years  ago  he  was  7  times  as 
old  as  A.     What  are  their  present  ages  ? 

6.  A  is  68  years  of  age,  and  B  is  11.  In  how  many  years 
will  A  be  4  times  as  old  as  B  ? 

Hint.     Let  N  equal  the  number  of  years.     Find  the  age  of  each  in  N" 
years,  and  then  form  the  equation. 


80  ALGEBRA 

7.  A  is  25  years  of  age  and  B  is  65.  How  many  years  ago 
was  B  6  times  as  old  as  A  ? 

8.  A  grocer  has  two  grades  of  tea,  a  60j^  grade  and  a  90^ 
grade.  He  wishes  to  make  a  mixture  which  he  can  sell  for 
80/  per  pound.  How  many  pounds  of  each  must  he  use  in  a 
mixture  of  120  pounds? 

Solution  :  1.  Let  n  =  the  number  of  pounds  of  60  ^  tea  used. 

2.  .-.  (120  —  n)  =  the  number  of  pounds  of  90^  tea  used. 

3.  .'.  60n  =  the  value  of  the  60 ^  tea  in  cents. 

4.  and  90(120  —  w)  =  the  value  of  the  90  ^  tea  in  cents. 

5.  .  •.  60  w  4-  90(120  —  n)  =  the  value  of  the  mixture  in  cents. 

6.  But  120  X  80  =  the  value  of  the  mixture  in  cents. 

7.  .-.  60  w  + 90(120  -  n)=  9600. 

8.  From  the  equation  n  =  40,  120  —  w  =  80. 
Check  :  40  pounds  of  tea  at  60/  are  valued  at  $24. 

80  pounds  of  tea  at  90  fi-  are  valued  at  f  72. 

Total  value  of  the  mixture  is  !$96. 

Also,  120  pounds  at  80  ^  are  valued  at  $  96. 

9.  A  grocer  has  tea  worth  70/  per  pound  and  other  tea 
worth  40/  per  pound.  How  many  pounds  of  each  must  he 
take  to  form  a  mixture  of  50  pounds  which  he  may  sell  at  49/ 
per  pound  ? 

10.  A  grocer  has  coffee  which  he  sells  at  36/  per  pound, 
and  other  coffee  which  he  sells  at  20  /  per  pound.  How  many 
pounds  of  each  must  he  take  to  make  a  mixture  of  100  pounds 
which  he  may  sell  at  25/  per  pound  ? 

11.  A  seedsman  wishes  to  make  a  mixture  of  grass  seed 
consisting  of  clover  seed  and  blue  grass  seed.  He  sells  his 
clover  seed  at  40/  per  pound,  and  his  blue  grass  seed  at  22/ 
per  pound.  How  many  pounds  of  each  must  he  take  to  make 
a  mixture  of  200  pounds  which  he  may  sell  for  25/  per  pound  ? 

12.  A  sum  of  money  amounting  to  $2.80  consists  of  dimes 
and  quarters.  The  number  of  dimes  exceeds  the  number  of 
quarters  by  7.     Find  the  number  of  each  kind  of  coin. 


MULTIPLICATION  81 

Solution  :  1.   Let      q  =  the  number  of  quarters. 

2.  Then  g  -j-  7  =  tlie  number  of  dimes. 

3.  .*.  25  g  =  the  number  of  cents  in  the  quarters, 
and                     \(i(q  -f  7)  =  the  number  of  cents  in  the  dimes. 

4.  .-.  25^4-10(^  +  7)  =280. 
Complete  the  solution  and  check  it. 

13.  A  man  has  two  kinds  of  money,  dimes  and  fifty -cent 
pieces.  If  he  is  oifered  $4.00  for  20  coins,  how  many  of  each 
kind  must  he  give? 

14.  A  sum  of  money  amounting  to  S2.20  consists  of  five- 
cent  pieces  and  quarters.  There  are  in  all  16  coins.  .  How 
many  are  there  of  each  kind? 

15.  A  sum  of  money  amounting  to  $24.90  consists  of  $2 
bills,  fifty-cent  pieces  and  dimes.  There  are  5  more  fifty-cent 
pieces  than  $2  bills,  and  3  times  as  many  dimes  as' $2  bills. 
How  many  are  there  of  each  denomination  ? 

61.   Equations  having  Fractional  Coefficients. 

Example  1.  If  the  sum  of  a  certain  number  and  one  half 
of  itself  be  diminished  by  three  iifths  of  the  number,  the  re- 
mainder is  9.     Find  the  number. 

Solution  :   1.    Let  x  —  the  number. 


(§10) 
(§57) 

(§9) 


10 +  5-6  =  9?    Yes. 
Note.    In  order  to  eliminate  the  denominators,  multiply  the  equation  by 
the  Lowest  Common  Multiple  of  the  denomiuators. 


Then 

-!-¥-• 

2.   The  denominators  must  be  eliminated. 

Mio: 

^     rr        2       0 
lOaJ  +  ie--  ~-^  '  —  =  90. 

3. 

4. 

10  a:  +  5  a;  -  6  a;  =  90. 

5.   Combining, 

9x  =  90. 

6.    D9: 

a;  =  10. 
5             2 

Check:  Does 

10  +  ^-3.^=9? 

(§10) 


82  ALGEBRA 

Example  2.     Solve  the  equation  ^-4^ =  — ^ . 

6       3       5       4 

Solution  :  1.    The  L.  C.  M.  of  6,  3,  5,  and  4  is  60. 
60(Lp-|)  =  60(?^i^-l). 

3.  70  m  -  100  =  30  m  -  15. 

4.  Aioo  :  Ss&m'  70  m  -  36  m  =  100  -  15.  (§  41) 

5.  .-.  34m  =  85;  w  =  ff  =  I  =2.5.    ' 

Check  :  This  solution  may  be  checked  by  substitution  or  by  going  over 
the  solution  again.  If  the  latter  method  is  used,  great  care  must  be  taken, 
as  it  is  easy  to  overlook  an  error. 

EXERCISE   40 
Solve  the  following  equations  : 
,     X  '  X     5  .- 

12. 

.•  ''-^      3 

14.   4d-^'  =  ''  +  ?3, 

5       2      20 

15     li_^'  =  ?_?i 
■     9        6      3      2' 


2     3      6 

m      7n  _  8 

2. 

3"      5  "15* 

3. 

-i-r 

4. 

¥=i«^ 

5. 

5a     3a      11 
3         4        6 

^ 

2r 
3  ' 

5r 
6 

=  6. 

m- 

3  ?/i  _  q 
'"5~'--' 

m 

2 

c 

5 

3c    ■ 

2c 

2 

3 

4 

3  * 

2  5  2  3 

7  "3 

—  —  ■[ 

5   ~      ■     4 

i  ic_4  X  _3 
14~T'"2 

:  ?/  _5 

8'~2 


7.  ?i  =  l4-l.  17. 

7       3^ 

8.  —  =  1+  —  .  18.    -^--_  — - 

9.  ^  =  i^-?.  19. 


10.    ^  +  il  =  ^.  20. 


22 

9 

52 

5 

3 

8 

6 

4* 

3.V 

_^_ 

.5?/ 

1 

2 

3 

4 

8* 

4^ 

-1  = 

7^ 

16 

9 

9  ' 

3  ' 

7?/i 

4  m 

11 

2m 

2 

~~3 

6 

5 

MULTIPLICATION  83 

EXERCISE  41 

1.  One  fifth  of  a  certain  number  exceeds  one  eighth  of  the 
same  number  by  3.     Find  the  number. 

2.  The  sum  of  three  numbers  is  65.  The  second  is  one 
half  of  the  first,  and  the  third  is  two  thirds  of  the  first.  Find 
the  numbers. 

3.  What  number  increased  by  one  half  of  itself  equals  the 
sum  of  two  thirds  of  itself  and  25  ? 

4.  What  number  exceeds  the  sum  of  its  third,  sixth,  and 
fourteenth  parts  by  18? 

5.  What  number  is  such  that  if  four  sevenths  of  it  be  sub- 
tracted from  itself,  the  result  equals  the  excess  of  three 
fourths  of  the  number  over  18? 

6.  What  number  is  such  that  if  two  thirds  of  it  be  in- 
creased by  100,  the  result  equals  four  fifths  of  it  ? 

7.  Seven  eighths  of  a  certain  number  is  as  much  less  than 
21  as  three  tenths  of  it  exceeds  2|.     W^hat  is  the  number  ? 

8.  The  difference  between  the  third  and  fifteenth  parts  of 
a  certain  number  is  28.     Find  the  number. 

9.  In  a  triangle  commonly  used  by  draughtsmen,  the 
second  angle  is  two  thirds  of  the  first,  and  the  third  angle  is 
one  half  of  the  second.     Find  the  angles  of  the  triangle.  (§  13) 

10.  In  another  triangle  used  by  draughtsmen,  there  are  two 
equal  angles,  each  of  which  is  one  half  of  the  third  angle. 
Find  the  angles  of  this  triangle. 

11.  There  are  three  consecutive  numbers  such  that  the  sum 
of  the  second  and  third  exceeds  three  halves  of  the  first  by  9. 
Find  the  three  numbers. 

12.  A  man  has  $4.35  in  dollars,  dimes,  and  cents.  He  has 
one  fourth  as  many  dollars  as  dimes,  and  five  times  as  many 
cents  as  dollars.     How  many  coins  of  each  kind  does  he  have  ? 

13.  The  Treasury  at  Washington  is  one  of  the  most  impos- 
ing of  the  Rational  buildings.     Its  perimeter  is  1400  feet.     Its 


84  ALGEBKA 

width  exceeds  one  half  of  its  length  by  25  feet.     Find  its 
dimensions. 

14.  The  greatest  depth  of  Lake  Superior  is  one  half  that  of 
Lake  Michigan ;  the  greatest  depth  of  Lake  Huron  exceeds 
one  sixth  that  of  Lake  Michigan  by  700  feet.  Tlie  depth  of 
Lake  Huron  exceeds  that  of  Lake  Superior  by  100  feet.  Find 
the  depth  of  each. 

15.  Probably  the  largest  room  in  the  world  under  one  roof 
is  the  passenger  concourse  of  the  Union  Station  in  Washing- 
ton, D.C.  Its  perimeter  is  1780  feet.  One  fifth  of  its  length 
exceeds  its  width  by  22  feet.     Find  its  dimensions. 

16.  Ten  times  the  population  of  the  United  States  in  1820, 
in  millions,  exceeded  the  population  in  1910  by  3.8  millions ; 
the  population  in  1910  exceeded  7  times  the  population  in 
1820  by  25  millions.     Find  the  population  in  both  years. 

17.  Plants  feed  upon  certain  plant  foods  present  in  the  soil, 
such  as  potash,  nitrogen,  and  phosphoric  acid.  A  fair  crop  of 
potatoes  will  remove  from  an  acre  of  ground  about  99  pounds 
of  these  three  foods.  The  amount  of  potash  removed  is  5 
times,  and  the  amount  of  nitrogen  2J  times  that  of  phosphoric 
acid.     Find  the  number  of  pounds  of  each  removed. 

18.  The  length  of  the  foundation  of  the  Capitol  in  Wash- 
ington exceeds  twice  the  width  by  51^  feet.  The  perimeter  of 
the  foundation  is  2202|  feet.  Find  the  dimensions  of  the 
foundations  of  the  Capitol. 

19.  The  average  wholesale  value  of  oak  lumber  in  1899  was 
$13.78  per  thousand  feet.  This  exceeded  one  half  of  the 
wholesale  value  in  1909  by  ,$3.53.  W^hat  was  the  wholesale 
value  per  thousand  in  1909  ? 

20.  The  distance  from  San  Francisco  to  London  via  New 
York  is  6990  miles.  The  part  of  the  journey  by  rail  is  50 
miles  less  than  |^  of  the  part  by  water.  Find  the  part  of  the 
journey  on  land  and  the  part  on  water. 


VI.  DIVISION 

62.  Division  is  the  process  of  finding  one  of  two  numbers 
when  their  product  and  the  other  number  are  given. 

To  divide  15  by  3  means  to  find  the  number  by  which  3  must  be  multi- 
plied to  give  the  product  15. 

The  Dividend  is  the  product  of  the  numbers ;  it  is  the  num- 
ber divided. 

The  Divisor  is  the  other  given  number ;  it  is  the  number  by 
which  the  dividend  is  divided. 

The  Quotient  is  the  required  number. 

63.  It  is  clear  that  a  -^  a  =  1 ;  for  d  x  1  =  a. 

64.  It  is  agreed  that  the  product \)f  zero  and  any  number  is 
zero.     This  makes  division  by  zero  impossible. 

Thus,  if  we  try  to  find  the  quotient  of  6  -v-  0,  and  let  q  equal  the  quo- 
tient, we  should  have  the  relation 

6  =  0  .  g. 

But  0  ■  q  =0  and  not  6,  so  there  cannot  be  any  ordinary  number  to  use 
as  q.     Hence,  there  is  no  number  to  represent  the  quotient  of  6  -f-  0. 

65.  Division  is  indicated  by  writing  a  fraction  whose  numera- 
tor is  the  dividend  and  whose  denominator  is  the  divisor. 

Thus,  the  quotient  of  15  h-  5  is  written  ^. 

The  quotient  of  7  a&c  -r-  3  xy  is  written  ^^-^  • 

Sxy 

Note.  The  line,  — ,  was  used  to  indicate  division  long  before  the  sym- 
bol, -r. 

66.  Division  of  a  Product  by  a  Number. 

Example.     Divide  6  x  8  by  2. 

Solution:!.  6x8--2=^-^. 

2 

85 


86  ALGEBRA 


2.  If  6  is  divided  by  2,         ^Ji3  =12^  =  24. 

2  )2 

4 

3.  If  8  is  divided  by  2,         1x8^6^^24. 

2  ^ 

4.  If  both  6  and  8  are  divided  by  2, 

3     4 

6x8^^x^^-^o 
2  ^ 

5.  Since  we  know  that     ^     =  48  h-  2  or  24,  it  is  clear  that  the  results 

2 

obtained  in  steps  2  and  3  are  correct,  but  that  the  result  in  step  4  is  in- 
correct.    Hence, 

Rule.  —  To  divide  the  product  of  two  or  more  numbers  by  a  num- 
ber, divide  any  one  of  the  factors  by  the  number,  but  divide  only  one 
of  them  by  it. 


EXERCISE 


42 


Find  each  of  the  following  indicated  divisions  in  two  ways: 

6 


1.    9x12  2     18x24^  3     28x56 


67.    The  Law  of  Signs  for  Division. 

Since  (-h  2)  x  (+  3)  =  +  6,  then  (+  6)  -f-  (-f  2)  =  +  3. 
Since  (- 2)x  (+3)  =  -  6,  then  (_  6) -- (-2)  =  +  3. 


Since  (-f  2)  x  (-  3)  =  -  6,  then  (-  6) 
Since  (- 2)  x  (-3)=-f  6,  then  (+6) 


(+2)=-3. 
(-2)  =  -3. 


If  the  signs  of  the  dividend,  the  divisor,  and  the  quotient  in 
each  of  the  previous  statements  are  examined,  the  following 
rules  become  clear : 

Rule.  —  1.  The  quotient  of  two  numbers  having  like  signs  is 
positive. 

2.  The  quotient  of  two  numbers  having  unlike  signs  is  negative. 


DIVISION  87 

EXERCISE  43 

1.  Divide  each  of  the  following  numbers  by  -t-3 : 

-hl2;        +15;        +27;        -18;        -36;        -42;        -57. 

2.  Divide  each  of  the  following  numbers  by  +  2  : 

_18;       +48;        +72;        -24;        -96;         +54;        -108. 

3.  Divide  each  of  the  numbers  in  Example  2  by  : 

(a)   -3;         (6)   +6;         (c)   -2;         (d)   -12. 

4.  Divide  each  of  the  numbers  in  Example  1  by  —  3. 

5.  Divide  each  of  the  numbers  in  Example  1  by  —  2. 

68.    The  Law  of  Exponents  for  Division. 

Development.  1.  Review  the  definitions  of  exponent,  base, 
and  power  of  a  number  in  §  15. 

2.  Divide  a^  by  a^. 

Ill 
Solution  :  \  =  ^-^ ^^— ^  =  1  .  a  •  a  =  a^. 

a°       jar.jgr.iir 

1  1   1 

Therefore       a^  -^  a^  =  a^.     Check  :  a^  .  a^  =  a^. 

Each  a  in  the  denominator  is  divided  into  one  of  the  a's  in  the  numerator. 
The  quotient  in  each  ease  is  1,  since  a  -^  a  =  1. 

3.  Find  as  in  step  2  the  following  quotients  and  write  the 
results  as  in  part  a : 

(a)  a^-7-a^  =  a^.  (c)  m^  h-  m^  =  ? 

4.  Examine  carefully  the  exponents  in  the  dividend,  the 
divisor,  and  the  quotient.  In  the  following  problems,  try  to 
give  the  results  immediately  without  going  through  the  solution 
as  in  step  2.     Test  by  multiplication. 

(a)  p'^p'=?  (c)  b'^-^b^=? 

(6)  a«-=-a'  =  ?  (d)  c^^c^=? 


88  ALGEBRA 

5.   Divide  a'^b^  by  a^b^. 

Ill         11 
SoLUTiox:  a^b^^a^b^=  ^^  =^'^'^' <^  '  ^ '^ ■  f>  ■  b  -  b  •  b  ^^^, 

11111 

Rule.  —  The  exponent  of  any  number  in  the  quotient  is  equal  to 
its  exponent  in  the  dividend  minus  its  exponent  in  the  divisor. 

Historical  Note.  — This  rule  was  known  to  Stifel  (see  note  §  53). 

DIVISION  OF  MONOMIALS  BY  MONOMIALS 

69.    Example  1.    Divide  -  14  a^b^  by  +  7  aK 

Solution  :  Use  the  Law  of  Signs,  §  67,  and  the  Law  of  Exponents, 

~  ^^  ^^^^  Z3  -  2  a(3-2)6--2  =  _  2  a62. 
+  7  a^ 

Check:     {+ 1  a^)  {- 2  ab'^)  =  -  U  a^b'^ . 
Example  2.     Divide  54  a^b-&  by  -  9  a'bh\ 

Solution  :       ^^  ^'^^^^  =-Q  a(7-4)5(2-2)c(3-2)  =  _  e  a%^c  =  -  6  aH. 
-  9  a*&^c2 

Check  :  (-  6  aH)  x  (-  9  a^b'^c'^)  =  +  54  d'bH^. 

Notice  that  by  the  law  of  exponents,  6^  _j_  ^2  _  ^2-2  _  ^o^ 

No  meaning  has  been  given  to  the  zero  poAver  of  a  number. 

Since  b^  -^  W  must  equal  1,  we  agree  that  W  =  1. 

Hie  zero  power  of  any  number  is  1.     Thus  : 
aO  =  l;  5«  =  1;  c'  =  l. 

Rule.  —  To  divide  a  monomial  by  a  monomial : 

1.  Make  the  quotient  positive,  if  the  dividend  and  divisor  have 
like  signs ;  make  it  negative,  if  they  have  unlike  signs. 

2.  Find  the  quotient  of  the  absolute  values  of  the  numerical  co- 
efficients. 

3.  Multiply  the  quotient  of  step  2  by  the  product  of  the  literal 
factors,  giving  each  its  exponent  in  the  dividend  minus  its  exponent 
in  the  divisor. 

4.  Omit  any  literal  factor  which  has  the  same  exponent  in  the 
dividend  and  divisor. 


DIVISION  89 

Example.     Divide  -  33  a%x^y^  by  +  3  aVy. 

Solution.     ( -  33  af^bx^y*)  -4-  (  +  3  a^x^y  )  =  -  1 1  aby^.     Ans. 

Check  :  These  solutions  may  be  checked  by  substitution,  for  they  must 
be  correct  for  all  values  of  the  literal  numbers  (except  0  sometimes) .  A 
better  way  is  to  use  the  rule  that  the  divisor  times  the  quotient  equals  the 
dividend. 

Here,  does  ( +  3  a^x^y)  x  (  -  1 1  aby^)  =  -  33  a^x^y*  ?    Yes. 

EXERCISE   44 
Divide : 

1.  oc^hja^.  21.  -  24  a*b^c  by  -  8  a*b\ 

2.  r'hyr*.  22.  2S  a^y^  by  -7  a^y. 

3.  y  by  J3».  23.  -  33  a^x'y*  by  -  3  ay. 

4.  m*  by  m\  24.  65  oi^y^^i^  by  — 13  xy^. 

5.  a^b^  by  ab.  25.  28  a^^^c®  by  - 14  a^bcl 

6.  7^9^  by  7^8.  26.  -  72  a^y  by  -  6  ^y. 

7.  x^y*  by  a^.  27.  -  40  a^6V  by  -  8  6c. 

8.  aW'  by  a^6.  28.  -  55  ar'/a;^  by  -  11  y-z^. 

9.  12  a*  by  2  a».  29.  -  70  aJb^c'  by  14  od^c. 

10.  15  a^  by  3  a^.  30.  -  96  m^n*  by  - 12  mw*. 

11.  20  ?V  by  4  r2.<?2.  31.  64:  ab^c  by  -4:  ab^c. 

12.  18  c-V^  by  9  cd^  32.  -  63  rV  by  7  r^s^. 

13.  —  14  win  by  —2  m.  33.  3  mn  by  —  6  m. 

14.  +  16  a^fe  by  -  4  a6.'  34.  -  5  r^s  by  15  r^. 

15.  —  10  a;y2  by  +5  ocy.  35.  4  a^y  by  —  ^  a;?/. 

16.  -2iyby~7p2.  36.  12  aft^^  by  -  24  afi^. 

17.  - 18  a^y  by  2  xy.  37.  10*  by  10^. 

18.  2  mV  by  wW  38.  2»  by  2\ 

19.  -  36  a«6<c2  by +  6 a*6V.  39.  3-' •  4^  by  3^ .  4^. 

20.  -  96  x'y^  by  - 16  ary.  40.  7»  •  8^  •  5  by  7^  •  5. 


90  ALGEBRA 

DIVISION   OF  POLYNOMIALS   BY  MONOMIALS 

70.    Development.     1.    Since  2  x  9  =  18,  then  i^  =  9. 

2.  Since2(a;  +  3)=2aj-f  6,  then  ?^i5  =  a;  +  3. 

id 

3.  Since  3  (a  —  5)  =  3  a  — 15,  what  does  ^^^ —  equal  ? 

4.  What  does  each  of  the  following  equal  ?     Test  the  result 
by  multiplying  the  divisor  and  quotient. 

5.  Since  a{h  ■\-  c)  —  ah  -\-  ac, 

then =  6  +  c. 

a 

Rule.  —  To  divide  a  polynomial  by  a  monomial : 

1.  Divide  each  term  of  the  dividend  by  the  divisor. 

2.  Unite  the  results  with  their  proper  signs. 

Example  1.     Divide  12  x^  —  &  x^ -\- "6  x  hj  —  3  «. 

Solution  :  — — — =  —  A  x^  +  2  x  —  \, 

—  3x 

Check  :      (-  4  a:^  +  2  a;  -  1)  •  (  -  8  x)  =  12  ic^  -  6  x^  +  3  a;. 

Example  2.     Divide  -9  a^  +  3  a^  _  12  a'  by  -  3  a\ 

Solution  :  {-12  a^  -  Q  a^  ^^  a^) -.  {- Z  a^)  =  Acfi  +  Z  a  -  1. 
Check  :  Multiply  the  quotient  by  the  divisor ;  the  result  should  equal 
the  dividend. 

EXERCISE  45 

Divide : 

1.  3  a- 6  6  by  3.  6.  -  3  a?*  +  6  a^  by  -  3  aj^. 

2.  16r-8.sby8.  7.  +  21  ?-^  -  14  r^s  by  -  7  rl 

3.  12  a^  -  IG  y"  by  4.  8.  18  m^n  —  27  mii^  by  9  ?mL 

■     4.   20  a2_  15  a  by  5  a.  9.  -  44  a^d  +  55  a^ft  by  11  «-6. 

5.    ^-x-  +  xhy  X.  10.  36  c^d^  -  48  c^d^  by  12  chK 


DIVISION  91 

11.  16a:»  +  28a;«-24ar  by  4ic2. 

12.  104  mV  —  52  m^n^  by  —  13  7nn.  - 

13.  6  a»6^c«  -  15  a«6V  -|-  3  a^6«c  by  -  3  a^b^c. 

14.  -  63  ar^/^*  -  84  a^y*z^  by  +  7  a^2/2^- 

15.  20  wi^i'  -  35  7n*n^  -  30  wi^w^  by  -  5  ??iV. 

16.  -  36  a"  +  108  a»  -  60  a"  by  -  12  a^ 

17.  32  a^b-c  -  24  ab'^c^  +  48  a6V  by  -  8  abc. 

18.  -63x-'«-18a;'-h45a:«-99aJ*by  -9.t*. 

19.  -  12  x'^j  -^-eaPy^-W  x'Y  +  20  xy'  by  -  2  an/. 

20.  60  a"  -  30  a^-  +  15  «^«  -  45  a^  by  -  15  a". 

21.  (a^6-a362^a268)--(a26). 

22.  (8x^-12a^z/)--(-4a:3)^ 

23.  (-21cV^'-42cd«)-(-7c(F). 

24.  (-6m3-h9?/l2_l2m)-^-(-3). 

25.  (_x*  +  ar^-l)-(-l)-'  -Vv    '    'V 

DIVISION  OF  POLYNOMIALS   BY  POLYNOMIALS 

71.   Division  of  a  polynomial  by  a  polynomial  is  like  long 
division  in  arithmetic. 

Divide  864  by  24.  3^ 

24  gii 


1. 

8d  -f-  24  =  3+. 

2. 

24  X  3  =  72  ;  subtract 

3. 

14 -i-  2  =  6+. 

4. 

24  X  6  =  144  :  subtract 

|72 

144 

144 


Divide  10  ar'  -  21  a^  -  11  a;  -f  12  by  2  a-  -  3  a;  -  4. 

5x-  3 

1.  10a;8-2x2  =  5x.  2x2-3x 

2.  (2x2-3x-4)x(5x);  subtract      .     .     . 


10a:«-21a;2-lla:+12 
10x8- 15x2 -20  a; 


3.  (-6x2) -(2x2)  =  -3.  -  6x2+  9X+12 

4.  (2x2-3x-4)x  (-3);  subtract      ....  -  6x^+  9x  +  12 
The  following  explanation  of  the  process  may  be  given. 


92  ALGEBRA 

We  are  to  find  an  expression  which,  when  multiplied  by  the  divisor, 
2  x^  —  S  X  —  4,  will  produce  the  dividend. 

The  term  containing  the  highest  power  of  x  in  the  product  is  the  prod- 
uct of  the  terms  containing  the  highest  powers  of  x  in  the  multiplicand 
and  multiplier. 

Therefore,  10  a^  is  the  product  of  2x'^  and  the  term  containing  the 
highest  power  of  x  in  the  quotient.  Dividing  10  x^  by  2  x^  gives  5  x, 
which  is  the  term  containing  the  highest  power  of  x  in  the  quotient. 

When  the  dividend  is  formed,  the  divisor  is  multiplied  by  each  term 
of  the  quotient,  and  the  results  are  added.  Now  reversing  the  process, 
multiply  the  divisor  by  5  x  and  subtract  the  result,  10  x^  —  15  a;^  —  20  x, 
from  the  dividend. 

The  remainder,  — 6  x^ +  9x4-12,  must  be  the  product  of  the  divisor 
and  the  rest  of  the  quotient.     Consider  it  a  new  dividend. 

Its  term  containing  the  highest  power  of  x,  —  6  x^,  is  the  product  of 
2  x^  and  the  term  containing  the  next  lower  power  of  x  in  the  quotient. 
Dividing  —  6  x^  by  2  x^  gives  the  next  term,  —  3.  Multiply  the  divisor 
by  —  3  and  subtract  the  result  from  the  previous  remainder.  There  is 
now  no  remainder. 

The  quotient  is  therefore  5  x  —  3. 

Rule.  —  To  divide  a  polynomial  by  a  polynomial : 

1.  Arrange  the  dividend  and  the  divisor  in  either  ascending  or  de- 
scending powers  of  some  common  letter. 

2.  Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor,  and  write  the  result  as  the  first  term  of  the  quotient. 

3.  Multiply  the  whole  divisor  by  the  first  term  of  the  quotient ; 
write  the  product  under  the  dividend  and  subtract  it  from  the 
dividend. 

4.  Consider  the  remainder  a  new  dividend,  and  repeat  steps  1,  2, 
and  3. 

Note  1.  The  terms  of  the  quotient  are  placed  above  the  tei-ms  of  the 
dividend  from  which  they  are  obtained. 

Note  2.    The  like  terms  are  carefully  arranged  in  a  vertical  column. 

Note  3.  Spaces  should  be  left  for  any  powers  of  the  common  letter 
which  are  not  present  in  the  dividend. 

Note  4.    As  in  arithmetic,  there  may  be  a  final  remainder. 

Example  1.     Divide  9  ab^-\-a^-9b^-5 a%  by  3  h^^-o?-2ah. 
Solution  :  1.   Arrange  according  to  descending  powers  of  a. 


DIVISION  93 

a -3  b 


a»-ba^b  +  dab'^-db» 
a»-2a^b-\-Hab^ 


2.  a'^^a^=a.  a'^-2ab-\-Sb^ 

3.  (a2_2a6+3ft2)x  a;  subtract     .  ...  

4.  -3a26--a2=-3  6.  -Sa^b+(iab-^-9b^ 
6.  (a2-2a6  +  362)x(-36);  subtract     ....  -Sa%-{-6ab'^-9b^ 

Check  :  Let  a  =  1  and  6  =  1. 
,  Divisor :  a- -2  ab -{- Sb^  =  1-2 -\- S  =  2. 

Dividend :  a^-5a'^b-\-9ab^-9b^  =  l-5  +  d-Q  =  -i. 
Dividend  -^  divisor :       (—  4)  -f-  (2)  =  —  2. 
Quotient :  a  —  S  b  =  I  —  S  =  —  2. 

Example  2.     Divide  xry^  -\- x*  —  y*  hy  —  xy -\- y'^  -{- a^. 

SOLDTION  : 

x2  _.  xy  +  y2 


— 

y" 

4-7fiy 

-  a:2y2  +  xy^ 

y' 

-xy^- 

-2t 
Note.    The  quotient  is  x^  +  xy  +  y^  ;  the  remainder  is  —  2  y^.     As  in 
arithmetic,  the  complete  quotient  may  be  written  : 

Complete  quotient :  x'^  +  xy  -{- y^  -] — — ^ — -  • 

x^—xy  +  y2 

Check  :  Let  x  =  1,  and  y  =  1.     Then,  dividend  =  1,  and  divisor  =  1. 
Quotient  =  1+  1  +  1  +  — ~^      =  34  —  =3-2  =  1. 

Since  1x1=1,  the  quotient  is  correct. 

Another  check  would  be  to  multiply  the  divisor  by  the  quotient  and 
add  the  remainder ;  the  sum  should  equal  the  dividend. 


EXERCISE  46 
Divide: 

1.  x^-\-5x  +  6hj  x-\-2.  4.   m^-\-Sm-\-12hym  +  2. 

2.  ar^+7a;  +  12bya;-f-4.  5.   A^+U  A-i-2Ahj  A-^3. 

3.  f^7y^i0hjy-\-5.  6.    r^  -  12  r  +  32  by  r  -  8. 


94  ALGEBRA 

7.  s2-13s  +  42  by  .s-7  11.    x"  -^  x —  6hy  x  -  2. 

8.  t2-{-6S-16thy  t-  9.  12.    t- -  t  -  SO  hj  t  +  5. 

9.  c2  +  72-17cby  c-8.  13.  a^- S  a-2S  hj  a-7. 
10.    d'-12d-\-36hy  d-6.         14.    m2-|-2  m- 15  by  m-f-5. 

15.  n'^-Tn^-SO  by  Ji^  +  S.  i 

16.  a^  -  17  a.x  +  60  a^  by  a:  -  5  a. 

17.  a2  +  5a&-6662by  a  +  116. 

18.  x^  —  2  xz  —  So  z~  hy  X  —  7  z. 

19.  x^ -[- 5  xry  —  24.  y^  by  a^  —  3  y. 

20.  x^y-  -  15  a.'?/  +  36  by  xy  -  3. 

21.  15a'2-llx-14by  3a.-  +  2. 

22.  6  a^  +  35  _  29  a  by  2  a  -  5. 

23.  12a2-28a  +  15by  6a-5. 

24.  30^2  + 8 -53  a:  by  6  0^-1. 

25.  32ar'-15/  +  28a;?/by  4a;  +  5y. 

26.  25  m^  +  40  mn  +  16  n^  by  5  ?/i  -f-  4  n. 

27.  a^-6x2-19i»4-84by  a;- 7. 

28.  6  a^  -  18  a  -  11  a^  +  20  by  2  a  -  5. 

29.  4  .T^  -  12  ?/3  +  17  xy^ -12x^yhy2x-S y. 

30.  12  a?  4-  6  a62  +  5  6'^  -  23  a^ft  by  4  a  -  5  h. 

31.  a;^  +  4  a.7/3  -|-  3  ?/*  by  a;^  —  2  a;?/  +  3  /. 

32.  2?i2_4_^5,^3_i9^^y  _g^^^5^2_3^ 

33.  12  +  13a^-19a;-12a^by  -3ar^-4  +  a;. 

34.  2a^H-8a-a3  +  15by  2a2-3a  +  5. 

35.  —  9  711^  —  16  +  m*  —  24  m  by  3  m  -|-  m^  +  4. 

36.  x'^  -\-  y^  -\-  Qi?y^  by  x^-^-y"^  —  xy. 


DIVISION  95 

37.  x-3 -I- 8  by  a; -h  2.  41.  x^ -\- f  by  x -\- y. 

38.  a^^  —  16  by  a;  —  2.  42.  ar"*  +  y*  by  a;  —  y. 

39.  ar*  —  ?/*  by  a;  -}-  ^y.  43.  ar'  +  32  by  a;  +  2. 

40.  a;*  —  2/^  by  a;  -  y.  44.  1  —  16  a<  by  1  +  2  a. 

45.  n^-16by  2n2  +  84-4w  fnl 

46.  13a:8H-71a; -70ar2-20  +  6a:*by  4  +  3ar^- 7aj. 
47".    n*  -h  4  m^n^  +  16  in*  by  2  inn'^  +  4  ?/i^  +  n*. 

48.  63a;^4-114ar^4-49ar'-16a;  +  20by  9  a;- -  5 -|- 6  a;. 

■  49.  ar^  +  50  -  70  a;  +  37  x^  by  10  -  2  x  +  x\ 

50.  10  a63  -  a'b-  -  25  6^  -f  16  a*  by  5  6^  +  4  a^  -  ab. 

51.  ar^  +  f  a;-lby  X-  +  2.  53.    6a:2_  5  ^_  1.  by  2a;- f 

52.  a^- Y-a^-f  1  bya:-^.        54.    ^a^^ -^'^ +  ^  by  ^a +3. 

Historical  Note. — Stifel  (1486-1567)  seems  to  have  been  one  of  the 
first  to  divide  a  polynomial  by  a  polynomial.  Sir  Isaac  Newton  (1642- 
1727),  in  a  book  published  in  1707,  pointed  out  the  advantage  of  arrang- 
ing the  dividend  and  divisor  according  to  ascending  or  descending  powers 
of  the  same  letter. 


VII.    SIMPLE   EQUATIONS 

72.  An  Equation  expresses  the  equality  of  two  numbers. 
Equations  are  of  two  kinds. 

73.  An  Identity  or  Identical  Equation  is  an  equation  whose 
members  are  equal  for  all  values  of  the  literal  numbers  in- 
volved ;  3iSy  3  x(a  —  b)  =  3 ax  —  S  bx. 

If        a  =  3,    b  =  \,   x  =  2,   3x(a-6)  =  3.  2(3- 1)=6.2  =  12; 
also,  3  aa: -3  &x  =  3. 3- 2-3. 1-2  =  18 -6  =  12. 

Any  other  set  of  values  of  a,  b,  and  x  will  produce  equal  numerical 
results  in  the  two  members  of  the  equation. 

An  identity  is  like  a  declarative  sentence;  it  makes  a  state- 
ment of  actual  equality. 

74.  An  equation  is  said  to  be  satisfied  by  a  set  of  values  of 
the  letters  involved  in  it  when,  after  substituting  these  values 
for  the  letters,  the  equation  becomes  an  identity. 

Thus,  xa  —  xb  =i  2 a  —  2b  'w,  satisfied  by  x  =  2,  for 

2a  —  26=2a  —  2?)isan  identity. 

a;  —  y  =  5  is  satisfied  by  a;  =  8,  y  =  3,  for 
8  —  3  =  5  is  an  identity. 

75.  A  Conditional  Equation  is  an  equation  involving  one  or 
more  literal  numbers,  which  is  not  satisfied  by  all  values  of 
the  literal  numbers. 

Thus,  {a)  X  +  2  =  5  is  not  satisfied  by  any  value  of  x  except  x  =  3. 
(6)  x2  —  5  a;  =  —  6  is  satisfied  by  x  =  2  and  by  x  =  3,  but  by  no  other 
values  of  X. 

A  conditional  equation  is  like  an  interrogative  sentence; 
it  implies  a  question. 

96 


SIMPLE  EQUATIONS  97 

Thus,  3  X  —  5  =  4,  asks  "  for  what  value  of  a;  is  3  a;  —  5  =  4  ?  " 
The  answer  is,  "  x  must  be  3,"  for  3  x  3  —  6  =  9  —  6  =  4. 

The  word  "  equation  "  usually  refers  to  a  conditional  equation. 

76.  If  an  equation  contains  only  one  unknown  number,  any 
value  of  the  unknown  number  which  satisfies  the  equation  is 
called  a  Root  of  the  equation.  . 

To  solve  an  equation  is  to  find  its  root  or  roots. 

Thus,  3  is  the  root  of  the  equation  x  +  2  =  5. 

77.  If  an  equation  has  only  one  unknown  number,  if  the 
unknown  does  not  appear  in  the  denominator  of  any  fraction, 
and  if  the  unknown  appears  only  with  the  exponent  1,  then 
the  equation  is  called  an  Equation  of  the  First  Degree,  or  a  Simple 
Equation. 

Thus,  3x  —  5  =  4  is  a  simple  equation. 

Historical  Note.  — The  idea  of  the  degree  of  an  equation  was  intro- 
duced by  Descartes. 

PROPERTIES  OF  EQUATIONS 

78.  Previously,  in  solving  equations,  four  rules  have  been 
employed : 

1.  The  same  number  may  be  added  to  both  members  of  an 
equation  without  destroying  the  equality.     (§  41.) 

2.  The  same  number  may  be  subtracted  from  both  members 
of  an  equation  without  destroying  the  equality.     (§  41.) 

3.  Both  members  of  an  equation  may  be  multiplied  by  the 
same  number  without  destroying  the  equality.     (§  10.) 

4.  Both  members  of  an  equation  may  be  divided  by  the 
same  number  without  destroying  the  equality.     (§  9.) 

All  simple  equations  are  solved  by  the  application  of  one  or 
more  of  these  rules.  However,  observation  of  the  results  of 
solving  equations  by  means  of  these  rules  leads  to  certain  more 
mechanical  methods  of  solution  which  may  be  used. 


98  ALGEBRA 

79.  Transposition.     Solve  the  equation  10  ic  —  5  =  3  07  -f  30. 
Solution  :  1.  10  x  -  5  =  3  a;  +  30. 

2.  A5:  10a;  =  3ic  +  30  +  5. 

3.  S&ci  10«-*3a:  =  30  +  5. 

4.  1x  =  35. 

5.  x  =  6. 

In  equation  3,  the  term  —  3  «  in  the  left  member  corresponds 
to  the  term  -j-  3  a;  in  the  right  member  of  equation  1 ;  and  the 
term  +  5  in  the  right  member  of  equation  3  corresponds  to 
the  term  —  5  in  the  left  member  of  equation  1.  These  are 
two  examples  of  transposition.  The  result  is  the  same  as  if  a 
term  were  taken  from  one  member  of  the  equation  and  placed 
in  the  other,  with  its  sign  changed. 

Rule.  —  A  term  may  be  transposed  from  one  member  of  an  equa- 
tion to  the  other,  provided  its  sign  is  changed. 

Historical  Note.  —  Our  word  a^gre&m,  curiously,  is  associated  with 
this  process,  transposition.  About  the  first  quarter  of  the  ninth  century 
an  Arabian  mathematician,  Mohammed  ben  Musa,  wrote  an  algebra,  for 
the  title  of  which  he  used,  Ilm  al-jabr  wa'l  muqabalah.  Al-jabr  meant 
the  process  of  transposing  terms.  This  title  was  used  in  various  forms 
in  Europe  until  about  the  fifteenth  century,  when  the  last  part  was 
dropped  and  algebra  came  into  use. 

The  Greeks  had  no  special  name  for  their  algebra.  The  Hindu  writers 
called  it  reckoning  with  unknowns. 

80.  Cancelling  Terms  in  an  Equation. 
Example.     Solve  the  equation  x-\-a  =  h  -\-a. 

Solution  :    x  -\-  a  =  h  -\-  a. 
Sa :  x  =  h. 

Thus,  the  term  a,  which  appeared  in  both  members  of  the 
given  equation,  does  not  appear  at  all  in  the  next  equation ; 
the  result  is  the  same  as  if  the  term  a  were  simply  dropped 
from  both  members. 

Rule.  —  If  the  same  term,  preceded  by  the  same  sign,  occurs  in 
both  members  of  an  equation,  it  may  be  cancelled. 


SIMPLE   EQUATIONS  99 

81 .   Changing  Signs  in  an  Equation. 
Example.     Solve  the  equation  a  —  x  =  b  —  c. 
Solution  :    1.        a  —  x  =  b  —  c. 
2.    M_i:  -a-\-x-—b  +  c. 

or  X  —  a  =  c  —  b. 

Thus,  the  signs  of  all  terms  of  the  equation  in  step  2  are  ex- 
actly opposite  to  the  signs  of  these  terms  in  the  equation  of 
step  1.  The  result  is  the  same  as  if  the  signs  of  all  the  terms 
of  the  equation  were  simply  changed. 

Rule.  —  The  signs  of  all  of  the  terms  of  an  equation  may  be 
changed,  without  destroying  the  equality. 

Note.  The  rules  given  in  §§  79,  80,  and  81  are  vahiable,  but  the  student 
should  endeavor  to  remember  that  they  arise  out  of  the  more  fundamental  rules 
given  in  §  78. 

Example.     Solve  the  equation 

7-5a;-9a;  =  15-9a;-3a;. 
Solution  :  1.       1  — 6x  — 9x  =  16—9x  — Sx. 

2.  Cancelling  the  term  —  9  a; :  7  -  5  x  =  15  —  3  a:.  (§80) 

3.  Transposing  -f  7  and  —Sx: 

-  5  a;  +  3  a;  =  15 -7.  (§79) 

-  2  a;  =  8. 

4.  Changing  the  signs  of  the  terms :  (§  81) 

2x=-8. 

5.  Dz  :  a:  =  -  4. 
Check  as  usual. 

EXERCISE  47 

Find  the  roots  of  the  following  equations ;  check  the 
solution  : 

1.  5a45  =  61-3a.  6.  13  +  4/)  =  lip -22. 

2.  9m-7  =  3m-37.  7.  5r- 12  =  16  -  9r. 

3.  13-6x=13a?-G.  8.  21-15z  =  -Sz -7. 

4.  7<-|- 10=16^-17.  9.  30  +  17c  =  27c-f  22. 
6.  15-6n  =  5?i-f48.  10.  19-162/ =  27  - 28 1/. 


100  ALGEBRA 

11.  2(5m  +  l)  +  16  =  44-3(m-7). 

12.  8^-5(4^  +  3)  =-3- 4(2^-7). 

13.  5c-6(3-4c)=c-7(4  +  c). 

14.  2(4a;+7)-6(2a;  +  3)  =  S(3ir-4)-7(2a;-3). 

15.  10r-(3r+2)=9r-(5r~4). 

16.  19-5c(4c  +  l)  =  40-10c(2c-l). 

17.  3_(a,._3)  =  5-2a;. 

18.  4(m  -  7)  =  5(m  +  10) -6(m  + 8). 

19.  2(r-l)  =  4(r-5)-3(r-2). 

20.  5  =  3(a;-2)-10(a;-6). 

82.  No  general  rule  can  be  given  for  the  solution  of  prob- 
lems.    The  following  suggestions  will  prove  helpful : 

1.  Every  problem  gives  a  relation  between  some  unknown 
numbers. 

2.  There  are  as  many  distinct  statements  as  there  are  un- 
known numbers. 

3.  Represent  one  of  the  unknown  numbers  by  a  letter ; 
then,  using  all  but  one  of  the  statements,  represent  the  other 
unknowns  in  terms  of  that  same  letter. 

4.  Using  the  remaining  statement,  form  an  equation. 

EXERCISE  48 

1.  Divide  44  into  two  parts  such  that  one  divided  by  the 
other  shall  give  2  as  the  quotient  and  5  as  the  remainder. 

Hint  :  The  dividend  =  divisor  x  quotient  +  remainder. 

2.  If  11  be  added  to  a  certain  number,  and  the  sum  be 
multiplied  by  5,  the  product  equals  —  6  times  the  number. 
Find  the  number. 

3.  Divide  19  into  two  parts  such  that  7  times  the  less  shall 
exceed  6  times  the  greater  by  3. 

4.  Divide  38  into  two  parts  such  that  twice  the  greater 
shall  be  less  by  22  than  5  times  the  less. 


SIMPLE  EQUATIONS  YOl 

6.  The  age  of  a  father  is  5  times  that  of  his  son  ;  his  age 
5  years  from  now  will  exceed  3  times  his  son's  age  by  4  years. 
Find  their  present  ages. 

6.  There  are  three  consecutive  odd  integers  such  that  when 
three  times  the  first  is  increased  by  the  second,  the  sum 
exceeds  3  times  the  third  by  5.     Find  the  numbers. 

7.  Divide  $  22  among  A,  B,  and  C  so  that  A  may  receive 
$  2.25  more  than  B  and  $  1.75  less  than  C. 

8.  Divide  49  into  two  parts  such  that  one  divided  by  the 
other  may  give  2  as  quotient  and  7  as  remainder. 

9.  Twice  the  width  of  the  Pennsylvania  Station  in  New 
York  exceeds  its  length  by  80  feet.  Four  times  the  length 
exceeds  the  perimeter  by  700  feet.     Find  the  dimensions. 

10.  Find  the  three  sides  of  a  triangle  if  the  perimeter  is 
45  inches,  if  the  second  side  is  twice  the  third  side,  and  if  the 
first  side  exceeds  the  third  by  5  inches. 

11.  Divide  134  into  two  parts  such  that  one  divided  by  the 
other  may  give  3  as  quotient  and  26  as  remainder. 

12.  The  elevation  of  Mt.  Whitney,  in  California,  the  highest 
point  recorded  in  the  United  States,  is  14,501  feet,  measured 
from  sea  level.  The  lowest  point  of  dry  land  in  the  United 
States  is  in  Death  Valley,  California.  If  52  times  the  elevation 
of  Death  Valley  be  diminished  by  45  and  the  result  be  in- 
creased by  the  elevation  of  Mt.  Whitney,  the  sum  is  zero. 
Find  and  interpret  the  elevation  of  Death  Valley. 

13.  A  now  has  one  third  as  much  money  as  B ;  after  B  gives 
him  $  24,  he  will  have  3  times  as  much  money  as  B  has  left. 
How  much  has  each  ? 

14.  A  cab  driver  finds  at  the  end  of  the  day  that  he  has 
$  11.55.  He  has  3  less  nickels  than  quarters,  twice  as  many 
half-dollars  as  quarters,  and  as  many  dimes  as  he  has  nickels 
and  quarters  together.  How  many  of  each  kind  of  coin  has 
he? 


102  ALGEBRA 

15.  A  gardener  decides  to  buy  $25  worth  of  gladiolus 
bulbs.  He  wants  some  of  the  pink  variety  which  sell  at  $  2 
a  hundred ;  two  thirds  as  many  of  the  yellow  variety,  at  $  3.50 
per  hundred,  as  of  the  pink  variety ;  and  four  times  as  many 
of  the  scarlet  variety,  at  $  1.50  per  hundred,  as  of  the  yellow 
variety.     How  many  of  each  shall  he  order  ? 

16.  In  an  isosceles  triangle,  two  sides  are 
equal  and,  also,  the  angles  opposite  these  sides 
are  equal.  Find  the  three  angles  of  an  isosce- 
les triangle  if  the  angle  between  the  equal  sides 
is  70°. 

17.  Find  the  sides  of  an  isosceles  triangle  if  its  perimeter  is 
720  inches  and  its  base  is  150  inches. 

18.  The  highest  temperature  recorded  in  the  United  States 
up  to  1907  was  119°,  recorded  in  Arizona.  The  lowest  tem- 
perature was  recorded  at  one  time  in  Montana.  If  twice  the 
lowest  temperature  be  decreased  by  9  and  the  result  be  added 
to  the  highest  temperature,  the  result  is  zero.  Find  and  inter- 
pret the  lowest  temperature. 

19.  The  area  of  Nebraska  exceeds  the  area  of  Virginia  by 
34,893  square  miles  ;  the  area  of  California  exceeds  three  times 
the  area  of  Virginia  by  30,416  square  miles ;  and  the  area  of 
California  exceeds  twice  the  area  of  Nebraska  by  3257  square 
miles.     Find  the  area  of  each  of  the  states. 

20.  In  1910,  the  total  number  of  boys  and  girls  in  the  public 
secondary  schools  was  915,061.  The  number  of  boys  exceeded 
three  fourths  of  the  number  of  girls  by  11,123.  Find  the 
number  of  boys  and  of  girls. 

21.  The  total  annual  income  from  two  investments  is  $250. 
One  sum  is  invested  at  4  %  and  the  other^  sum,  which  exceeds 
the  first  by  $  500,  is  invested  at  5  % .  Find  each  of  the  sums 
invested. 

Solution  :  1.  Let  s  =  the  smaller  sum  in  dollars. 

2.  .'.  .04  s  =  the  interest  on  this  sum. 


SIMPLE   EQUATIONS  108 

3.  .'.  (s  -I-  600)  =  the  larger  sum  in  dollars. 

4.  .-.  .05(s  +  500)  =  the  interest  on  this  sum. 
6.    .-.  .04  8  +  .05  (s  +  500)  =  250. 

6.  .-.  .04  s  +  .05  s  +  25  =  250. 

7.  .-.  .09  s  =  225. 

8.  .-.  s  =  2500. 

9.  .-.  s  -I-  500  =  3000. 

Check  :  5  %  of  $  3000  =  §  150  ;  4  %  of  .9  2500  =  $  100  ; 
and  $  150  +  $  100  =  .^  250. 

22.  One  sum  of  money  is  invested  at  5  %  ;  a  second  sum  is 
invested  at  G  %.  If  3  times  the  first  sum  exceeds  the  second 
sum  by  $  100,  and  if  the  total  income  is  $  155,  find  the  sums 
invested. 

23.  A  man  has  $5000  invested  at  4  %.  How  much  money 
must  he  invest  at  6  %  to  make  the  total  income  equivalent  to 
5  %  on  the  total  amount  invested  ? 

24.  A  man  has  $  3000  invested  at  3.5  %,  and  S  4500  at  4  %. 
How  much  must  he  invest  at  6  %  to  make  the  total  income 
equivalent  to  5  %  on  the  total  sum  invested  ? 

25.  A  man  owns  a  number  of  shares  of  U.  S.  Steel  Preferred 
Stock  ($  1000  par  value)  which  pay  7  %  annually,  and 
5  times  as  many  bonds  of  the  Chicago  Edison  Company, 
($1000  par  value)  which  pay  5%.  If  his  total  income  is 
S  960,  how  much  has  he  invested  in  each  form  ? 

83.  Distance,  Rate,  and  Time  Problems.  —  If  a  train  goes  a  dis- 
tance of  240  miles  in  6  hours,  it  travels  at  an  average  rate  of 
40  miles  per  hour. 

The  time  (t)  is  expressed  as  a  number  of  units  of  time ;  as 
hours,  minutes,  days. 

The  rate  (r)  is  expressed  as  a  number  of  units  of  distance 
covered  in  the  unit  of  time  ;  as,  a  number  of  miles  per  hour,  or 
a  number  of  feet  per  second,  etc. 


104  ALGEBRA 

The  distance  (d)  is  expressed  as  a  number  of  units  of  distance 
covered  in  the  total  time. 

From  the  example  and  the  definitions,  it  is  clear  that : 

the  distance  equals  the  rate  multiplied  by  the  time. 

d=rt.  (1) 

From  equation  (1),  -  =  r  or  r=  -;  that  is 

the  rate  equals  the  distance  divided  by  the  time. 

From  the  equation  (1),  -  =  t  or  t  =  --,  that  is, 
r  r 

the  time  equals  the  distance  divided  by  the  rate; 

thus  the  time  occupied  in  going  200  miles  at  40  miles  per  hour 
is  5  hours. 

EXERCISE  49 

1.  Express  the  distance  covered  by  a  train  in  15  hours  at  the 
rate  of : 

(a)  5  miles  per  hour;  (c)   (x-\-7)  miles  per  hour; 

(b)  R  miles  per  hour ;  (d)  (2  y  —  3)  miles  per  hour. 

2.  Express  the  distance  covered  by  a  train  in  ^  hours  at  the 
rate  of: 

(a)  m  miles  per  hour ;  (b)  (x  -j-  9)  miles  per  hour. 

3.  Express  the  time  required  by  an  automobile  to  go  a  dis- 
tance of  300  miles  at  the  rate  of : 

(a)  30  miles  per  hour ;  (c)  {x  -\-  5)  miles  per  day  ; 

(b)  n  miles  per  hour  ;  (d)  (m  —  4)  miles  per  day. 

4.  Express  the  time  for  a  trip  of  N  miles  at  the  rate  of  : 
(a)  10  miles  per  hour ;  (b)  x  miles  per  hour. 

5.  At  what  rate  does  a  man  travel  who  goes  250  miles : 

(a)  in  10  days ;  (c)  in  (x  —  5)  hours  ; 

(b)  in  Qi  days ;  (d)  in  (r  +  7)  days. 


SIMPLE  EQUATIONS 


105 


6.  At  what  rate  does  a  man  travel  who  goes  D  miles : 
(a)  in  16  hours ;     (6)  in  t  days ;     (c)  in  {x  —  4)  minutes. 

7.  The  rate  of  one  train  is  r  miles  per  hour.  Express  the 
rate  of  a  train  which  travels  5  miles  more  per  hour. 

8.  Express  the  distance  traveled  by  each  of  the  trains  in 
Example  7  in  15  hours. 

9.  Suppose  that  the  distance  gone  by  the  second  train 
exceeds  that  gone  by  the  first  train  by  25  miles.  Form  an 
equation  expressing  this  fact. 

10.  A  man  on  foot  and  a  man  on  a  bicycle  both  travel  for 
5  hours,  the  rate  of  the  latter  exceeding  that  of  the  former  by 
7  miles  per  hour.     Let  r  represent  the  rate  of  the  former. 

(a)  Express  the  rate  of  the  second  man. 

(6)  Express  the  distance  each  travels. 

(c)  Form  an  equation  expressing  the  fact  that  the  sum  of 
the  distances  is  60  miles. 

Eqvxttions 

In  the  following  problems,  express  the  time,  rate,  and  dis- 
tance traveled  by  each  party,  and  then  form  the  equation 
from  the  given  relations.  It  is  usually  wise  to  illustrate  the 
problems  geometrically. 

11.  Two  men  travel  toward  each  other  from  points  which 
are  150  miles  apart  at  rates  of  5  and  15  miles  an  hour  respec- 
tively.    In  how  many  hours  will  they  meet? 

Solution  :  1 .  Let  h  =  the  number  of  hours  until  they  meet. 


Then  for 

the  time  is 

the  rate  is 

the  distance 

one  man 

h  hours 

6  m.  an  hr. 

6  h  mUes 

the  other  man 

h  hours 

15  m.  an  hr. 

15  h  miles 

3.   Since  the  sum  of  the  distances  is  150  miles,  b  h  +  16h  =  150, 

5h     _  I5h 


ISO 


106  ALGEBRA 

12.  Suppose  that  the  more  rapid  traveler  starts  two  hours 
after  the  other  in  Problem  11.     When  will  they  meet  ? 

13.  Suppose  that  two  men,  who  travel  at  the  rate  of  6  miles 
and  10  miles  per  hour  respectively,  start  from  the  same  place 
in  opposite  directions.  In  how  many  hours  will  they  be  200 
miles  apart  ? 

14.  Suppose  that  A,  traveling  10  miles  per  hour,  leaves  a 

place  3  hours  before  B;  suppose  that  B  travels  15  miles  per 

hour.     In  how  many  hours  will  B  overtake  A  ? 

>^ — i I 

B D|  Hint  :  A  is  at  C  when  B  starts ;   B  overtakes  A 

at/>. 

15.  Suppose  A,  traveling  15  miles  per  hour,  starts  4  hours 
before  B.  At  what  rate  must  B  travel  to  overtake  A  in  10 
hours  ? 

16.  Two  hours  after  A  left,  B  starts  after  him  in  an  auto- 
mobile at  the  rate  of  27  miles  an  hour  and  overtakes  him  in  2i 
hours.     At  what  rate  was  A  traveling  ? 

17.  A  and  B  travel  toward  each  other  from  points  separated 
by  250  miles,  A  at  a  rate  wljich  exceeds  B's  rate  by  8  miles 
an  hour.  If  they  meet  in  5  hours,  at  what  rate  did  each 
travel  ? 

18.  Some  boys  who  are  boating  on  a  river  know  that  they 
can  go  with  the  current  6  miles  per  hour  and  can  return  against 
the  current  at  the  rate  of  3  miles  per  hour.  How  far  may 
they  go  if  they  have  only  3  hours  for  the  trip  ? 

19.  A  man  has  11  hours  at  his  disposal.  How  far  may  he 
go  in  a  buggy  at  the  rate  of  10  miles  an  hour  if  he  plans  to 
return  at  an  average  rate  of  7  miles  per  hour  ? 

20.  An  automobile  is  traveling  at  the  rate  of  25  miles  an 
hour.  In  how  many  hours  will  a  second  automobile  overtake 
the  first  if  the  second  starts  2  hours  later  than  the  first,  and 
travels  at  the  rate  of  35  miles  an  hour  ? 


SIMPLE   EQUATIONS  107 

21.  An  express  train  whose  rate  is  36  utiles  an  hour  starts 
54  minutes  after  a  slow  train  and  overtakes  it  in  1  hour  and 
48  minutes.     What  is  the  rate  of  the  slow  train? 

22.  An  automobile  party  is  traveling  at  the  rate  of  20  miles 
per  hour.  At  what  rate  must  a  second  automobile  travel  in 
order  to  overtake  the  first  if  it  starts  2  hours  after  the  first  and 
wishes  to  overtake  it  in  3  hours  ? 

23.  Chicago,  and  Madison,  Wisconsin  are  about  140  miles 
apart.  Suppose  that  a  train  starts  from  each  city  toward  the 
other,  one  at  the  rate  of  35  miles  per  hour  and  the  other  at 
the  rate  of  40  miles  per  hour.     How  soon  will  they  meet? 

84.  Problems  about  Thermometers.  There  are  two  kinds  of 
thermometers  in  common  use,  the  Fahrenheit  and  the  Centigrade. 
The  Fahrenheit  is  the,  one  with  which  most  of  us  are  familiar. 
The  Centigrade  is  used  by  scientists  throughout  the  world. 

It  is  necessary  at  times  to  change  a  temperature  reading  on 
one    scale   to    the    corresponding   reading  on  the 
other  scale. 

The  temperature  at  which  water  boils  and  that 
at  which  it  freezes  are  called  the  "boiling"  and     ioo° 
the  "freezing"  points.     On  the  Fahrenheit  scale 
these  points  are  marked  212°  and  32° ;  on  the  Centi- 
grade scale,  100°  and  0°  respectively.     The  num- 
ber of  Fahrenheit  degrees  between  these  two  points 
is  180,  and  the  number  of  Centigrade  is  100.    Hence,       t 
100  Centigrade  degrees  correspond  to  180  Fahren-       S 
heit  degrees,  or  1  Centigrade  degree  to  f  Fahrenheit 
degree. 

17.8 

Note.  This  does  not  mean  that  a  temperature  of  1°  C. 
is  the  same  as  f°  F.  1°  C.  is  one  degree  above  0;  the 
corresponding  Fahrenheit  reading  is  §°  above  32  (the  freez- 
ing point),  or  33|^.  Thus,  a  temperature  of  1°  C.  =  a  tem- 
perature of  SSI*^  F. 


f\    ^ 


II 


108  ALGEBRA 

EXERCISE  50 

1.  How  many  Fahrenheit  degrees  are  equal  to  the  following 
number  of. Centigrade  degrees? 

(a)  15;         (b)  25;         (c)  50;         (ri)  100. 

2.  Remembering  that  Centigrade  degrees  above  freezing  are 
counted  from  zero,  and  Fahrenheit  from  32,  what  Fahrenheit 
temperature  corresponds  to  the  following  Centigrade  tempera- 
ture? 

(a)   +15°C. 

Solution  :  1.   15  Centigrade  degrees  =  27  Fahrenheit  degrees. 

2.  15°  C.  above  freezing  =  27°  F.  above  32  =  59°  F. 
.'.  15°  C.  corresponds  to  59°  F. 

(b)   +30°C.;         (c)   +55'^C.;         (d)  - 10°  C. 

3.  Derive  a  formula  for  changing  Centigrade  temperature 
readings  into  Fahrenheit  readings. 

Solution  :  1.   Let       C°  =  the  Centigrade  reading. 
Let  F°  =  the  Fahrenheit  reading. 

2.  C  Centigrade  degrees  =(f  C)  Fahrenheit  degrees. 

3.  C°  counted  from  0,  the  Centigrade  freezing  point,  =(§  C)°  counted 
from  32,  the  Fahrenheit  freezing  point. 

.-.  i^  =  32  +  f  C. 
Check  :  Let  C  =  0.         .-.  i^  =  32  +  §  •  0  =  32. 
Let  C  =  100.     .-.  if  =  32  +  1 .  100  =  32  +  180  =  212. 

Since  the  freezing  and  boiling  temperatures  correspond,  the  solution  is 
correct. 

4.  The  formula  can  be  used  to  change  Fahrenheit  into  Centi- 
grade readings. 

Change  —  13°  F.  to  Centigrade. 

Solution  :  1.  -  13  =  32  +  |  C.     (Substituting  in  the  formula.) 

2.  .-.   -  65  =  160  +  9  C. 

3.  .-.- 225  =  9(7,  or  C=- 25°; 

i.e.  25°  below  zero  Centigrade. 


SIMPLE   EQUATIONS  109 

6.  In  Physics  and  Chemistry,  the  temperature  —  273°  C. 
is  important.  To  what  Fahrenheit  temperature  does  this  cor- 
respond ?     (Substitute  in  the  formula.) 

6.  The  following  substances  melt  at  the  temperatures  indi- 
cated.   To  what  Fahrenheit  temperatures  do  these  correspond  ? 

ParaflBn  +  55°    C.  Iron  +  1200°  C. 

Tin  +  232°  C.  Mercury  -  39°  C. 

7.  Attempts  have  been  made  to  get  record-breaking  low 
temperatures.  The  following  table  gives  low  temperatures 
produced,  the  name  of  the  experimenter,  and  the  date  of  the 
experiment.  To  what  Fahrenheit  temperatures  do  these 
correspond  ?  • 

Experimenter 

Fahrenheit 

Faraday 

Dewar 

Onnes 

8.  The  temperatures  at  three  places  in  the  United  States 
on  a  certain  day  were : 

(a)   -foO°F.;         (b)   -f- 12°  F. ;         (c)   -  8°  F. 
What  would  these  temperatures  be  on  a  Centigrade  scale? 

9.  The  following  liquids  boil  at  the  temperatures  indicated : 

Alcohol  172.4°  F.         Turpentine  320°  F. 
Give  the  boiling  temperatures  on  the  Centigrade  scale. 

10.  Air  can  be  liquefied  by  reducing  its  temperature  until 
it  reaches  — 182°  C.  To  what  Fahrenheit  temperature  does 
this  correspond  ? 


Datx 

Temperatttbe 

1714 

-17°C. 

1823 

-  102°  C. 

1898 

-262°C. 
-  269°  C. 

1908 

VIII.     SPECIAL    PRODUCTS   AND   FACTORING 

85.  In  arithmetic,  it  is  found  necessary  to  memorize  the 
multiplication  table  as  an  aid  in  multiplication,  division,  and 
factoring.  In  algebra,  also,  certain  forms  of  number  expres- 
sions occup  frequently,  which  must  be  multiplied,  divided,  or 
factored  by  inspection. 

86.  To  Factor  an  algebraic  expression  is  to  find  two  or  more 
expressions  which  will  produce  the  given  expression  when  they 
are  multiplied  together. 

Review  the  definitions  of  factor  (§  28)  and  common  factor 

(§  11)-  * 

87.  A  number  which  has  no  factors  except  itself  and  unity 
is  called  a  Prime  Number ;  as,  3,  a,  and  x-\-y. 

A  monomial  is  expressed  in  items  of  its  prime  factors  thus  : 

12  a^hh  =  2  '2'3'a'a'a'b'b'C. 

88.  Squaring  a  Monomial. 

Development.     1.    What  does  x^  mean  ?     (xyY?     (2r^sy? 

2.  Find  each  of  the  following  squares  by  multiplication  : 

(a)  (2xyy;         (b)  (Sa^by-,         (c)  (- 2  .^s^. 

3.  Compare  the  exponent  of  each  letter  of  the  square  with 
the  exponent  of  that  letter  in  the  given  monomial. 

Rule.  —  To  square  a  monomial : 

Square  its  numerical  coefficient,  and  multiply  the  result  by  each 
of  the  literal  factors  of  the  monomial,  giving  each  letter  twice  its 
original  exponent. 

110 


SPECIAL   PRODUCTS  AND   FACTORING 


111 


89.    Cubing  a  Monomial. 

Development.  1.  Find  each  of  the  following  cubes  by 
multiplication : 

(a)  {2x^yf;        (b)  (3  7-V)^         (c)  (-2^fy. 

2.  Compare  the  exponent  of  each  letter  of  the  cube  with 
the  exponent  of  that  letter  in  the  given  monomial. 

Rule.  —  To  cube  a  monomial : 

Cube  its  numerical  coefficient,  making  the  result  negative  if  the 
given  monomial  is  negative,  and  multiply  the  result  by  the  literal 
factors  of  the  monomial,  giving  each  letter  three  times  its  original 
exponent. 

Example  1.  Find  (-5  xYf. 
Solution  :  (  —  5  x^y^)^  =4-25  rr^j/^. 

Example  2.  Find  (  -  5  x^ff. 
Solution  :  (  -  5  x^)^  =  -  125  ofiy^. 

EXERCISE  51 

1.  What  sign  does  the  square  of  any  number  have  ? 

2.  What  sign  does  the  cube  of  a  negative  number  have  ? 

3.  Learn  thoroughly  the  squares  of  the  numbers  from  1  to  20. 

4.  Learn  thoroughly  the  cubes  of  the  numbers  from  1  to  6. 


Give  the  values  of  the  fol 

lowing  indicated  powers  : 

5. 

(a^bf. 

15. 

(-IxYf. 

25. 

(-18)1 

6. 

(-a'b'y. 

16. 

(-{-Sa'bJ. 

26. 

(-i6ty. 

7. 

(2a^f)'. 

17. 

(-5mhiy. 

27. 

(-\-Gmny. 

8. 

(-{-abcf. 

18. 

(-  9  r^sty-. 

28. 

a«)^. 

9. 

(+  2  a'bf. 

19. 

(+4(r>d2)l 

29. 

(imnf. 

10. 

(-^Sxyzf- 

20. 

(11  ab'cy. 

30. 

(-in^f. 

11. 

(-5  my. 

21. 

(-  12  m'ny. 

31. 

(+fa^6)l 

12. 

{-2x^y)\ 

22. 

(-\-5m*ny. 

32. 

(-icdy- 

13. 

(-3x*yy. 

23. 

(-15v*zy. 

33. 

(-i^yy- 

14. 

i-e^fy. 

24. 

{- 10  c'dy. 

34. 

(-i^sy. 

112  ALGEBRA 

90.  The  Square  Root  of  a  Monomial.  If  an  expression  can  be 
resolved  into  two  equal  factors,  it  is  said  to  be  a  Perfect  Square, 
and  one  of  the  factors  is  said  to  be  its  Square  Root. 

Thus,  4  a%^  is  equal  to  2ab^  x2  ab^  ;  hence  it  is  a  perfect  square  and 
2  ab^  is  its  square  root. 

Note.  4  a^b^  is  also  equal  to  (—  2  ab^)  x  (—  2  ab^) ,  so  that  —  2  ab^  is  also  a 
square  root.    In  this  chapter,  only  the  positive  square  root  will  be  considered. 

The  following  questions  lead  to  the  rule  for  extracting  the 
square  root  of  a  perfect  square  monomial. 

Development.  1.  What  sign  does  the  square  of  any  mo- 
nomial have  ? 

2.  When  squaring  a  monomial,  what  do  you  do  with  the  ex- 
ponents of  the  literal  factors  ?   with  the  coefficient  ? 

3.  In  finding  the  square  root,  then,  what  should  you  do  with 
the  exponents  of  the  literal  factors  ?   with  the  coefficient  ? 

4.  Find  the  square  root  of  each  of  the  following  monomials, 
and  test  the  result  by  multiplication  : 

(a)  a^;  (b)  Ax'y^;  (c)16r2s^;  (d)  25  a^yh\ 

Rule. --1.  A  perfect  square  monomial  is  positive,  has  a  perfect 
square  numerical  coefficient,  and  only  even  numbers  as  exponents. 

2.  To  find  its  square  root :  find  the  square  root  of  its  numerical 
coefficient,  and  multiply  the  result  by  the  literal  factors  of  the  mo- 
nomial, giving  each  letter  one  half  of  its  original  exponent. 

The  symbol  for  extracting  the  square  root  is  the  Radical  Sign, 
V ;  the  vinculum  is  usually  combined  with  it,  V  . 

Example.     Find  the  square  root  of  25  m%®. 
Solution  :    \/25  m^n^  =  5  mhi^. 

91.  The  Cube  Root  of  a  Monomial.  If  an  expression  can  be 
resolved  into  three  equal  factors,  it  is  said  to  be  a  Perfect  Cube, 
and  one  of  the  factors  is  said  to  be  its  Cube  Root. 

Thus,  since  27  a%^  is  equal  to  3  a^ft  •  3  a^b  •  3  a^ft,  it  is  a  perfect  cube, 
and  3  a^b  is  its  cube  root. 


SPECIAL  PRODUCTS   AND   FACTORING  113 

The  following  questions  lead  to  the  rule  for  extracting  the 
cube  root  of  a  perfect  cube  monomial. 

Development.  1.  What  sign  does  the  cube  of  a  positive 
number  have  ?   of  a  negative  number  ? 

2.  When  cubing  a  monomial,  what  do  you  do  with  the  ex- 
ponents of  the  literal  factors  ?   with  the  coefficient  ? 

3.  In  finding  the  cube  root,  then,  what  should  you  do  with 
the  exponents  of  the  literal  factors  of  the  monomial?  with  the 
coefficient  ? 

4.  Find  the  cube  root  of  each  of  the  following  monomials 
and  test  the  result  by  multiplication  : 

(a)  a^  (b)  a^«;  (c)  f;  (d)  mV; 

(e)  8  a' ;  (/)  27  m« ;        (g)   -  c« ;  (h)   -  64  ^f. 

Rule.  —  1.  A  perfect  cube  monomial  has  a  perfect  cube  numerical 
coefficient,  whose  sign  may  be  +  or  -,  and  its  literal  factors  have 
exponents  which  are  exactly  divisible  by  3. 

2.  To  find  its  cube  root :  find  the  cube  root  of  the  numerical  co- 
efficient, making  it  positive  or  negative,  according  as  the  sign  of  the 
monomial  is  +  or  -  ;  and  multiply  the  result  by  the  literal  factors 
of  the  monomial,  giving  each  letter  one  third  of  its  original  exponent. 

The  symbol  for  extracting  the  cube  root  is  the  radical  sign 
with  the  Index  3,  as  follows  :   -y/  . 

Example.     Find  the  cube  root  of  —  125  aW. 


Solution  :    V  -  125  a%^  =  -  5  a^b^. 

EXERCISE  52 
Find  the  indicated  roots : 
1     V4m^  6.    -^27mV.  11.    ■\/-64:a^y\ 


2.  V9m«.  7.    V36a2fe«.  12.    V225  a^y . 

3.  V25  d'b\  8.    -^-Sa'b^  13.    V2567¥. 


4.    Va«6^  9.    V-125m«ni2.      14.    V+216c3dV. 


5.    V8a36«.  10.    V169mV.  15.    V-27r3p. 


114  ALGEBRA 


16.    </+125Ii^                     /4^y-  27.    </n?. 

. ,  22.    \ ^ 

17.  Vi«i  ^'25c'  28.  ^j;^;m. 


23.    aV  1  . 


2T- 


19.    V-216^¥.  24.    Via^ftl 


20.    ■\/3f-c'd\  25.    V2^5mV.  ^   49  c^ 

21     a/~^  26     aK  31.    J2M^. 

92.  It  is  not  always  possible  to  factor  a  polynomial.  Those  • 
polynomials  which  can  be  factored  are  the  products  of  certain  J 
special  forms  of  number  expressions. 

CASE  I 

Type  Form :  a{h  +  c)  ==  a6  +  ac. 

93.  Multiplication.     The  rule  for  multiplying  a  polynomial 
by  a  monomial  is  given  in  §  57. 

EXERCISE  53 
Find : 

1.  2x'{x'-3xy  +  y^).  4.    +  5  a{3  x"  -  2  xy -\- y"). 

2.  —'dxyioi^  +  xy  —  'f).  5.    a6(3  a^  — 2  a&  +  &^). 

3.  H-3mn(m'^  — m^w  +  Ti^).        6.    —oa{2a—h  +  c). 

94.  Factoring  a    Polynomial   whose  Terms    have   a   Common 
Monomial  Factor. 

Example  1.     Factor  ax-f-3a  —  ha. 

Solution  :  1.  Each  term   has  the  factor   a.     Divide  the  expression 
by  a. 

2.    Then  ax  +  ^  a  -  ha  =  a{x +  ^  -  h). 

Check  :  a  •  {x  -\-  Z  —  b)  =  ax  -\-  ^  a  —  ah. 

Example  2.     Factor  14  xy^  —  35  a^?/l 

Solution  :  1.     Each  term  contains  the  factor  7  xy^. 

2.   Dividing  the  expression  by  7  xy^,  the  quotient  is  2  ?/2  _  5  x^. 


SPECIAL   PRODUCTS   AND   FACTORING  115 

3.    Whence         U  xy^  -  35  .rV  =  7  xy2(2  y'^  -br^). 
Check  :  7  xy^{2  y'^  -bx^)  =  14  xy^  -  35  xf^y^ 

Rule.  —  To  factor  a  polynomial  whose  terms  have  a  common  mo- 
nomial factor : 

1.  Find  the  greatest  common  factor  of  its  terms. 

2.  Divide  the  polynomial  by  it. 

3.  The  factors  are  the  common  factor  found  in  step  1,  and  the 
quotient  obtained  in  step  2. 

EXERCISE  64 
Factor  the  following  polynomials : 

1.  Sx-\-  3  y.  11.   2  m''  -h  4  mn  +  2  71^. 

2.  4  m  —  4  a.  12.   x^y  +  xy^  +  y^. 

3.  2r-6s.  13.   4  a^  -  H  ab  +  4:b\ 

4.  5xt-10xs.  14.   3x2-3a;?/  +  3  2/^ 

5.  Sax'-2ay^.  15.    r^a^  -  2  r^a;?/ +  ? V- 

6.  2  rm»- 16  rw^.  16.    12  a^-20  a»  +  4  a^. 

7.  a^  +  4rc.  17.    3a:2_i5a;_|_i8. 

8.  3m*— 6m.  18.    a/'  —  5ar^  +  6a. 

9.  a;*  -  af».  19.   49  m¥  -  16  ?i¥. 
10.   30  r^- 5?^.                           20.    15  6--6  6\ 

21.  4:  a^b^- 20  a¥x-{- 25  b^x". 

22.  -2«25  +  i6a.6-32  6. 

23.  3?iy-21n2/3  +  18/. 

24.  5  aay^y  —  5  axy  —  30  a?/. 

25.  4  a6ar^  + 16  afta;?/- 20  a6/. 

26.  9m2<-6m^-63f. 

27.  48a;y-144a:3^  +  108a;y. 

28.  ar^  —  a;^H-a:^  —  ar^. 

29.  3  am^  —  6  amn  +  3  an~  —  3  ap^. 

30.  5aa^  —  15  oaz-^y  + 15  axy^  —  5  ai/^. 


116 


ALGEBRA 


31.    Recall  that  the  area  of  a  triangle  is  ^  of  the  product 
of  its  base  and  altitude.     Thus,  the  area  a 

ofAABC=^a'b.  (§  17) 

Indicate  the  area  of  a  triangle  of  base 
m  and  altitude  p. 


32.  Suppose  that  the  polygon  ABODE F  can  be  divided 
into  six  triangles,  such  that  their  altitudes  are  all  equal.  Call 
the  altitudes  each  a,  and  the  bases  b,  c,  d,  e^ 
f,  and  g. 

(a)  What  is  the  area  of  A  OBC?  A  ODC? 
A  ODE  ?  etc.  ? 

(b)  Indicate  the  sum  of  these  areas. 

(c)  Simplify  that   sum  by  removing  the 
monomial  factor. 

(d)  Simplify  the  result  by  substituting  p  for 

(b+c  +  d-^e-\-f+g). 
The  final  result  should  be :  area  =  ^  ap. 

33.  Suppose  that  the  altitude  of  A  RXT  is  a,  and  the  alti- 
tude of   A  EST  is  c.     The  base  of   each 

is  6.  fff 

(a)  Represent  the  area  of  each. 

(b)  Indicate  the  sum  of  these  areas.  /  /a 

(c)  Simplify  the  result  by  removing  the  ;f« ^ai^- 

monomial  factor. 

34.  The  area  of  a  circle  whose  radius  is  r  is  ttt^. 

(a)  What  is  the  area  of  the  circle  of  radius  E  ? 

(b)  How  can  you  find  the  area  of  the  ring 
between  the  large  and  small  circles  ?  Indicate 
this  area. 

(c)  Simplify  the  result  of  step  b  by  removing 
the  monomial  factor. 

(d)  Find  the  value  of  the  result  when  M  is  5 
and  r  is  4. 


SPECIAL   PRODUCTS  AND  FACTORING  117 

35.    Suppose   that,  in   the   adjoining   figure,  the  rectangles 
have  equal  bases  of  length  m,  and  altitudes 
of  length  a,  b,  c,  etc. 

(a)  Represent  the  area  of  each. 

(b)  Indicate  the  sum  of  these  areas. 

(c)  Simplify  the  resulting  expression  by 
removing  the  monomial  factor. 


^rbha 


m    m  m    m    m   m    m 


CASE   II 
Type  Form:  (a -f  &)- =  or^  +  2  fl6  +  6^. 

95.   The  Square  of  a  Binomial. 

Development.     1.   What  does  (a -|- 5)^  mean  ? 

2.  Find  the  value  of  the  following  by  actual  multiplication, 
as  in  §  58,  and  write  the  results  as  in  part  (a) : 

(a)  (a-|-6)2  =  a2  +  12a+36.  (c)  (m  +  5y=? 

(b)  (6  +  4)2  =  ?  (rf)  (0^  +  8)2=  ? 

3.  Observe  carefully  the  results  in  step  2.  Then  try  to 
find  the  following  squares  mentally,  first,  and  check  by 
multiplication : 

(a)  (x-  +  2)2=?  (c)  (z  +  7y  =  ? 

(b)  (2/ +  3)2=?  (d)  (A:  +  10y-'  =  ? 

4.  Write  the  sum  of  x  and  y.  Indicate  the  square  of  that 
sum.  Find  the  value  of  the  square  either  mentally  or  by 
multiplication. 

5.  Prove  by  multiplication  the  following  fact : 

(a  +  6)2  =  a2  +  2a6  +  6l 

Rule.  —  To  square  the  sum  of  two  numbers : 
Square  the  first  number ;  add  twice  the  product  of  the  two  num- 
bers ;  add  the  square  of  the  second  number. 

Example.     Square  (3  a +  2  6c). 
Solution  :  (3  a  +  2  bey  =  (3  ay  +  2(3  a)  (2  he)  +  (2  bey 
=  9  a2  +  12  a6c  +  4  feV. 


118 


ALGEBRA 


The  solution  inay  be  checked  by  substitution,  but  it  is  necessary  to 
acquire  such  skill  in  doing  these  problems  that  checking  in  that  manner 
will  be  unnecessary. 

Rule.  —  To  square  the  difference  of  two  numbers  : 
Square  the  first  number ;   subtract  twice  the  product  of  the  two 
numbers ;  add  the  square  of  the  second  number. 

Example.     Square  (4  a^  —  5  b^). 

Solution  :  (4  a2  -  5  b^y  =  (4  a^y  -  2(4  a'^){b  b^)  +  (5  b^y 

=  16  a*  -  40  a%^  +  26  ¥. 
Note.     In  actual  practice,  pupils  should  do  all  of  this  work  mentally, 
passing  from  the  given  problem  to  the  result  as  follows  : 
(3  m  -  5  w)2  =  9  m2  -  30  mn  +  25  n^. 
This  is  called  "  finding  the  result  by  inspection.'''' 
The  following  figure  illustrates  the  square  of  (a-\-b). 


a 

ab 
a 

b          b^ 

a2 

a          ab 

1  ab         4-        ab 

a2  + 


(a4-&)-=a2  +  2a6  + 


EXERCISE  55 
Square  the  following  binomials  by  inspection : 


1. 

a +  5. 

8. 

.a^-10. 

15. 

m-\-4:n. 

2. 

6  +  6. 

9. 

7-2  + 12. 

16. 

2p-Sq. 

3. 

c-7. 

10. 

mn  — 11. 

17. 

x^-Sy. 

4. 

d-S. 

11. 

2a  +  b. 

18. 

2x-\-5. 

5. 

m^-^4.. 

12. 

3a-c. 

19. 

5m- 6. 

6. 

7l'  -  8. 

13. 

a-^2b. 

20. 

3a'-2b\ 

7. 

i>^  +  V 

14. 

r  —  3s. 

21. 

2xy-{-9. 

SPECIAL  PRODUCTS   AND   FACTORING  119 

22.  3a^-6b.  25.    9  a^  +  o  r\  28.    2xy-9z\ 

23.  10  r  -f  4  t-.  26.    7  —  2  al  29.    7  a6  —  5  cd. 

24.  lls-5<.  27.    8c  +  3mV.  30.    9a*-h()b^ 

Note.  For  additional  drill  problems,  if  desired,  square  a  binomial  like 
3  a: +  6,  making  b  successively  1,  -,  3,  etc.  up  to  10.  Then  change  3  to  4  or 
any  other  number.  Short  daily  drills  of  this  sort  afford  good  mental 
arithmetic. 

Expand  the  following : 

31.  (m-if.  34.  (x  +  iy.  37.  (r  +  i/. 

32.  (y-if.  35.  (n  +  if.  38.  (s -\- ^  tf. 

33.  (2-h|)^  36.  (p-iy.  39.  (x-iyy. 

40.~  Square  29  mentally. 

Solution  :  292  =  (30  -  1)2  =  900  -  60  +  1  =  901  -  60  =  841 . 

This  should  be  done  mentally. 

41.  Square  32.      (Hint  :  32  =  30  +  2). 
Square  mentally  the  following  numbers : 

42.  21.  46.    32.  50.    29.  54.   52. 

43.  22.  47.    33.  51.   28.  55.   43. 

44.  23.  48.    19.  52.   39.  56.    57. 

45.  31.  49.    18.  53.    38. 

57.  Problem.  Find  a  rule  for  squaring  any  number  ending 
in  5. 

Solution  :  1.    35  =  3  x  10  +  5  ;  45  =  4  x  10  +  6  ;  56  =  5  x  10  +  5. 

2.  Similarly,  any  number  ending  in  5  may  be  represented  by 

10  n  +  5. 
Thus,  for  95,  n  is  9,  since  9  x  10  +  5  =  95. 

3.  (10  n  -\-  5)2  =  100  w2  4-  100  w  +  25 

=  100n(n+  l)  +  26, 
or,  n  ■  (n  +  1)  hundreds  +  25. 

Thus  the  square  of  95,  in  which  n  =  9,  is  . 

9  .  (9  +  1)  hundreds  +  25,  or  9025. 


120  ALGEBRA 

Rule.  —  To  square  a  number  ending  in  5,  drop  the  5,  multiply  the 
balance  of  the  number  by  the  consecutive  integer,  and  affix  25  to  the 
result. 

Example.     85^  =  7225. 

I.e.  8  X  9  =  72 ;  affixing  25,  the  result  is  7225. 

58.  Find  by  this  rule  the  squares  of  some  numbers  ending 
in  5,  such  as  35,  45,  105,  115,  etc. 

96..  Factoring  Perfect  Square  Trinomials.  In  algebra,  it  is 
necessary  to  be  able  to  recognize  a  perfect  square  trinomial. 

Development.  1.  Square  the  following  binomials,  and 
write  the  result  as  in  part  (a)  r 

(a)  (a  +  6)2  =  a^  +  2  a6  +  h\  (c)     (3  o^  +  4  yf  =  ? 

(b)  (2  a  +  3  6)2  =  ?  (d)  (4  m  -  5  n)'  =  ? 

2.  How  many  terms  are  there  always  in  the  square  ? 

3.  What  sign  does  the  first  term  of  the  square  have  ?  the 
third  term  ? 

4.  Notice  that  the  first  and  third  terms  are  perfect  squares 
and  that  the  second  term  is  twice  the  product  of  the  square 
roots  of  these  two  terms. 

5.  Are  the  following  perfect  squares  ?  Give  the  reason  for 
your  opinion.     Give  the  square  roots  of  the  perfect  squares : 

(a)  c'-{-2cd-\-d\  (c)   r^-6r  +  9. 

(6)  m^  +  2  7nn  -\-  n\  (d)  a^  _  10  a;  -  25. 

Rule.  —  1.  A  trinomial  is  a  perfect  square  when  two  of  its  terms 
are  perfect  squares  and  positive,  and  when  the  remaining  term  is 
twice  the  product  of  the  square  roots  of  the  perfect  square  terms. 

2.  To  find  the  square  root  of  a  perfect  square  trinomial :  extract 
the  square  roots  of  the  two  perfect  square  terms,  and  connect  them 
by  the  sign  of  the  remaining  term. 

Example  1.     Is  4  ar^  +  9  ^/'^  —  1 2  a;/  a  perfect  square  ? 

Solution  :  4  cc^  is  a  perfect  square  ;  its  square  root  is  2  x. 
9y^  is  a  perfect  square  ;  its  square  root  is  3  y^. 
12x2/2  =  2(2a;)(3j/2). 


SPECIAL   PRODUCTS  AND   FACTORING  121 

Hence  4  x^  +  9  y^  —  12  xxf-  is  a  perfect  square,  and  its  square  root  is 

2  X  -  3  y2. 

Example  2.     Is  9  a^  +  7  a6  +  4  ^-  a  perfect  square  ? 
Solution  :  9  a^  and  4  6^  are  perfect  squares.    Their  square  roots  are 

3  a  and  2  6,  respectively. 

2(3  a)  (2  6)  =  12  ah.  Since  the  third  term  is  7  aft,  and  not  12  a6, 
9  a2  _).  7  (jft  _(.  4  52  ig  not  a  perfect  square. 

To  be  a  perfect  square,  the  term  7  ah  would  need  to  be  changed  to  -|- 
or  —  12  ah. 

Then  V9  a2  i  12  aF+TP  =  3  a  ±  2  6. 

EXERCISE   56 

Supply  the  missing  term  so  as  to  make  perfect  square  tri- 
nomials of  the  following  expressions,  and  then  give  the  square 
roots  : 

1.  ar  +  (?)  4-  in^'  H.  >«''  +  4  m  +  (?)•      " 

2.  a2  -  (?)  ^i\  12.  a^  -  6  a;  4-  (?)• 

3.  m^  -  (?)  H-  n2.  13.  f  -  12  ?/  +  (?)• 

4.  ^6  _  (?)  ^_  ^2.  14.  z'^  -  10  2^  +  (?). 

5.  a;<  _  (?)  4.  9 2/2.  15.  9a2  4-6a  +  (?). 

6.  9  ar^  +  (?)  +  4  /.  16.  144  A^  -  (?)  -f  25. 

7.  If)  r<  -  (?)  +  25  «2.  17.  9  62  -  (?)  +  36  c}. 

8.  100  m2  -f  (?)  +  4  n".  18.  25  x'^  +  (?)  -f  36  f. 

9.  25  c-'- (?)  + 9(^2.  19.  49  r^  -  (?)  +  25  s2. 
10.    81  c«  -  (?)  -f  25  d\  20.  9  a;2  _  (?)  ^_  64. 

In  the  following,  determine  whether  the  trinomials  are  per- 
fect squares ;  find  the  factors  when  possible. 

21.    4m2-  20  7mr  4- 25  ?i^ 

Solution  :  1.  This  is  a  perfect  square,  according  to  the  rule.  (§  96, 
Rule  1.) 

2.    Hence,     4  m^  -  20  mn'^  +  25  n*  =  (2  m  -  6  it^)^ 

=  (2m-  5n2)(2m-6n2). 


122  ALGEBRA 

22.  m^  —  10  mn  +  25  n\  34.  169  m^  -  26  m^n  -f-  w^. 

23.  .T^  +  12  ^2^  +  36 /.  35.  QA  a'b- -{- U  abed -\- c'd^. 

24.  mV  4- 18  maj  -f  81.  36.  100  x^  -  80  a:^  +  16. 

25.  64  a^  +  15  ab  +  61  37.  49  ^4  _|_  112  m-'  +  64. 

26.  100x'-60xhj^9i/.  38.  9  a- +  42  «6  +  49  52. 

27.  49  a;-/- 70  aj?/;^ +  25  22.  39.  121  a^ft-^  +  130  a6c  +  36  c^. 

28.  4  a^  -  22  ax  +  25  a^.  40.  64  a'  +  176  «&  +  121  b\ 

29.  81  X-  +  16  ?/  —  72  xy.  41.  a^^  +  a.-  +  J. 

30.  4a2-28aa.'  +  49a^.  42.  y^-^-^y  +  i- 

31.  25  x^  -I-  16  2/2  —  40  a:?/.  43.  z^  +  /^  «^^  —  I  ^w- 

32.  9  mV+  25  r^-  30  7)i?i?'2  44  ^  ^^,2  ^  25  ^i^  _  i_o  rnn. 
33  4  ^4  _^  36  a;2  -  12  a-^2  45  _4^  ^2  _^  x2_  ^y  _^  g  y2^ 

97.  Complete  Factoring.  When  a  number  is  to  be  factored, 
all  of  its  prime  factors  should  be  found.  The  factors  found 
first  may  sometimes  be  factored  again. 

Thus,  48  =  8.6  =  4.2-2.3=:2.22.2.3. 

In  algebra,  this  sort  of  factoring  is  frequently  necessary. 

Example  1.     Find  the  prime  factors  of 
5  0771^  —  50  amn  + 125  aiv^. 

Solution  :  5  am^  —  50  amn  +  125  a7i^ 

=  5a(m2- 10m/i  + 25^2)  (§94) 

=  5  a  (m  -  5  n)  (m  -  5  n).  (§  96) 

Do  not  fail  to  rewrite  all  factors,  like  the  5  a,  which  are  not  factored 
again. 

Rule.  —  To  find  the  prime  factors  of  an  expression : 

1.  First  remove  any  monomial  factor  which  may  be  present. 

2.  Then  factor  the  resulting  expression,  when  possible,  rewriting 
all  expressions  which  cannot  be  factored. 


SPECIAL   PRODUCTS   AND   FACTORING  123 

EXERCISE  57 
Find  all  of  the  prime  factors  : 

1.  3mx'-^24:7nx-\-ASm.  6.    20  a^  -  20  a^bd" -t  5  b'c*. 

2.  lSt~12at  +  2aH.  7.   30 a^b - 120 ab-^  120b. 

3.  7  m-n^i- 70  mn'-{-175n\  8.   3aa^ -Saxy -\-3ay'. 

4.  5  x^yh-\- 70  xyz-^  215  z.  9.   75cx' +  IS  cy^ -120  cxy. 
6.   11  mx'y^-^  22  7nxy-j-U  711.  10.   5  7^s  - 10  rst -h  20  st\ 

CASE  III 

Type  Form :     (a -{-b){a  —  b)=a^-  Ir. 

98.    The  Product  of  the  Sum  and  the  Difference  of  Two  Numbers. 
Development.     1.    Find   by   multiplication    the    following 
products,  and  write  the  results  as  in  part  (a)  : 

(a)  (a;-f-3)(.r-3)  =  a^-9.  (c)    (A;  + 10)  (A;  -  10)  =  ? 

(6)   (m  +  7)(m-7)=?  {d)  (r  +  9)(r-9)  =  ? 

2.  Observe  the  results  in  step  1 ;  try  to  find  the  following 
products  mentally.     Check  the  results  by  multiplication. 

(a)  (a  +  6)(a-6)  =  ?  (c)    (d  +  4)(d-4)  =  ? 

(6)   (c-h8)(c-8)  =  ?  id)  (2/  +  5)(y-5)  =  ? 

3.  Write  the  sum  of  x  and  y ;  write  the^ir  difference.     Find 
the  product  of  the  results. 

4.  Prove  by  multiplication  the  following  fact : 

(^a  +  h){a-b)=a'-b\ 

Rule.  —  To  find  the  product  of  the  sum  and  the  diiference  of  two 
numbers : 

1.  Square  each  of  the  numbers. 

2.  Subtract  the  second  square  from  the  first. 

Example  1.     Find  (5  a^  -\-  m)  (5  a^  —  m). 

Solution  :   (6  a^  +  in)  (5  a^  -  m)  =  (6  a^)^  -  {my  =  25  a*  -  wi^. 

Example  2.     Find  mentally  the  product  of  24  and  16. 
Solution  :  24  x  16  =  (20  +  4)  (20  -  4)  =  400  -  16  =  384. 


124  ALGEBRA 

EXERCISE  58 

Find  by  inspection  the  products : 


1. 

(a +  2)  (a -2). 

11. 

(10  0^2/ -11)  (10  0^2/ +  11). 

2. 

(r-3)(r  +  3). 

12. 

(13m2-12)(13m2  +  12). 

3. 

(s^  +  4)(s2-4). 

13. 

(a'-h')(a'-\-b'). 

4. 

(x^-^5y)(x^-5y). 

14. 

(^~f)(^-^f). 

5. 

(3  m  +  4  7i)  (3  7n  —  4  w). 

15. 

(aj«-8/)(a^  +  8/). 

6. 

(m^-l)(m^  +  l). 

16. 

(7a:/^-10)(7a;?/2^  +  10). 

7. 

(z-3d)(z-{-3a). 

17. 

(|^-*)(l^  +  i)- 

8. 

(Sk'-9t){SJc'  +  9t). 

18. 

(im  +  |)(|m-|). 

9. 

(Sab-7c){3ab-\-7c). 

19. 

(I^^-A)(^'^+A). 

10. 

(4r-5s2)(4r  +  5s2). 

20. 

an^-i)(in'  +  i). 

21. 

(9 +  5)  (9 -5).             25. 

32 

28.            29.   55-65. 

22. 

(25 +  2)  (25 -2).         26. 

53 

47.             30.   33-37. 

23. 

22  .  18.                          27- 

62 

58.             31.    41-49. 

24. 

23  .  17.                          28. 

98 

.102.           32.    22-28. 

33.  Find  the  cost  of  18  dozen  of  eggs  at  22^  per  dozen. 

34.  Find  the  cost  of  16  yards  of  gingham  at  24^  per  yard. 

35.  Find  the  cost  of  45  yards  of  scrim  at  55  j^  per  yard. 

99.   Factoring  the  Difference  of  Two  Perfect  Squares. 
Development.     1.   What  is    the   product   of    (a +  2)   and 
(a  —  2) ?     What,  then,  are  the  factors  of  a^  —  4? 

2.  Find  the  factors  of  : 

(a)  a2-9;  (c)  k^-P; 

(b)  m'-W;  {d)  9r2-4s2. 

3.  Write   the   square  of   r;    of   t;  the   difference  of   these 
squares.     Factor  the  result. 

4.  Similarly,  the  factors  a^  —  b^  are  : 

(a'-b')  =  (a  +  b){a-by 


SPECIAL   PRODUCTS  AND  FACTORING  125 

Rule.  —  To  factor  the  difference  of  two  squares  : 

1.  Find  the  square  root  of  the  two  perfect  square  terms. 

2.  One  factor  is  the  sum  of  the  results ;  the  other  factor  is  the 
difference  of  the  results. 

Example  1.     Find  the  factors  of  25  r*  -  16  tf^. 

Solution  :  25  r*  -  16  <«  =  (5  r^)*  -  (4  t^y^ 

=  (5r2  4-4«8)(5r-2_4«8). 

Example  2.     Find  mentally  the  value  of  13^  —  71 
Solution  ;  132  -  72  =  (13  +  7)  (13  -  7) 

=  20  X  6  =  120. 
This  example  shows  how  the  above  rule  can  be  used  to  simplify  arith- 
metical work. 

EXERCISE  59 

Factor  the  following  token  possible : 


1. 

a*  -  b\ 

12. 

36  -  49  y'. 

23.    100a262c2-l. 

2. 

(f-9. 

13. 

1  -  36  a262. 

24.    c*-|dl 

3. 

16 -dl 

14. 

o^-i. 

25.    81a;i0-196y«r 

4. 

a^-1. 

15. 

100  2V -49. 

26.    256n^-m2. 

5. 

l-y\ 

16. 

/'-im2. 

27.    225-64^2.   ^ 

6. 

x^-4.y\ 

17. 

ia^-i^<^- 

28.   85a^-y2 

7. 
8. 

9 -ml 

18. 
19. 

64m^-81n«. 
169  a2- 196. 

29-  -^-i- 

a^     d^ 

9. 

16^^-25/. 

20. 

25  a?  - 1. 

30.    ?5-^. 

10. 

81r2-16  2^ 

21. 

144  x^  - 121 2/2. 

r2      36 

11. 

25a«-8l2«. 

22. 

^^>^-A- 

Find  mentally  the 

following : 

31. 

162-91 

36.    352 

-152. 

32. 

232-72. 

37.    272 

- 132. 

33. 

242  -  62. 

38.    262 

-42. 

34. 

332-172. 

39.    952 

-52. 

35. 

242-162. 

40.    752 

-252. 

126  ALGEBRA 

Find  the  prime  factors  of  the  following : 

41.  5  x^  -  5  2/2.     (See  §  94.)     46.   36  xv^  -  xiA 

42.  mH  —  2ot.  47.    ttR-  —  tts\ 

43.  Zai"" -12  as\  48.    \  irdr  -  \ -rrm'. 

44.  4c-c?— 9d^  49.    m^  —  n\ 

45.  32?/  — 2  05^.  50.    o^-y^. 

Find  the  following  quotients  : 

51.  ix'-y'-)-^{x-y).  55.  (4  s^  -  ^^^ -- (2  s  -  ^). 

52.  {a''-w})-^{a  +  m).  56.  (9x-2-16  ?/2)--(3x-|-4  ?/). 

53.  (r2-9)-(r-3).  57.  (2om'-Un'')^{^m  +  4.n). 

54.  (^2  _  25)  -  (?  +  5).  58.  (169  -  100  a^)- (13 -10  a). 

Tell  what  binomial  will  divide  each  of  the  following ;  give 
the  quotient : 

59.  9cc2-4  2/*.  62.    144/ -121. 

60.  m^- 16  7*2.  63.    256  c* -400. 
ea.   25ci«-36  6^  64.   100r2-36i^ 

EXERCISE   60 

Review 
Expand  the  following : 

1.  (x'-y-y.  4.    2pg(^''-s')(r^  +  ^). 

2.  2a(x-\-yXx-y).  5.    3c(2a-6)l 

3.  3m{c(^-y%a^-\-y^.  6.    5ab{a^-by. 

Factor  completely  the  following  : 

7.  r*-2rV  +  s^  10.    3  ax^  -  6  axY  +  S  ay*. 

8.  mC*  —  2  mc^  4-  m.  11.5  mx-  —  10  mxy  -\-  5  my^. 

9.  5tx^-5ty\  12.    2  r^m*  -  4  ?^m V -f  2  ?''V. 


SPECIAL   PRODUCTS  AND   FACTORING  127 

CASE  IV 

Type  Form :  (x  +  a){x  ^  h)  =  x-  -{- (a -\-  h)x  +  ab. 
100.    The  Product  of  Two  Binomials  having  a  Common  Term. 

Development.     1.    Find  by   actual  multiplication  the  fol- 
lowing products,  and  write  the  results  as  in  part  (a) : 

(a)  (a;-|-2)(a;  +  3)=ir2  4- 5x-f  6. 

(6)  (x  4-  5)(a:  -f  3)  =  ?  {d)  (m  -  7)(m  -  2)  =  ? 

(c)  (a-\-b){a-^Q>)  =  ?  (e)  (s-5)(s-8)=? 

2.    Observe   carefully   the   results  in   1.     Try   to   find  the 

following  products  mentally;  check  the  results  by  multipli- 
cation : 

(a)  (6  +  4)(6  +  2).  (c)  (x-3)(a;-7). 

(6)  (cH-4)(c-f3).  (cZ)  (y-4)(2/-5). 

Rule.  —  To  obtain  the  product  of  two  binomials  having  a  common 
term: 

1.  Square  the  common  term. 

2.  Multiply  the  common  term  by  the  algebraic  sum  of  the  second 
terms  of  the  binomials. 

3.  Find  the  product  of  the  second  terms. 

4.  Add  the  results. 

Example  1.   Find  the  product  of  a;  —  8  and  a;  -|-  5. 
Solution:  (x  -  8)(x  +  5)  =  a;2+(- 8  +  6)ic+(-8)(+  5) 
=  a;2  -  3  a;  -  40. 

Example  2.   Find  {ah  +  2){ab  -  11). 

Solution  :  {ah  +  2)  {ah  -  11)  =  a^"^  -  9  a6  -  22. 

Note.    Here,  one  glances  at  +  2  and  —  11,  notes  that  their  sum  is  —  9  and 
that  their  product  is  —  22,  and  writes  the  result  as  above. 

Example  3.     Find  mentally  the  product  of  23  and  24. 
Solution  :  1.    23  x  24  =  (20  +  3)  (20  +  4)  =  400  +  7  •  20  +  12 

=  400  +  140  +  12  =  662. 


58 

ALGEBRA 

EXERCISE   61 

Find  the  following  products  : 

1. 

(a;  +  2)(a^  +  3). 

26. 

(rs-St)(rs  —  St). 

2. 

(x-\-2)(x  +  A). 

27. 

(a  +  3&)(a  +  136). 

3. 

(a +  3)  (a +  5). 

28. 

(c  -  4.  d)(c -12  d). 

4. 

(a+4)(a  +  6). 

29. 

(x^-Sy){x^-10y). 

5. 

(m  +  5)(m  +  9). 

30. 

(x'-^5f)(a^-2f). 

6. 

(b-S)(b-7). 

31. 

(p^  + 11  q)(p^^  15  q) 

7. 

(5_4)(&-8). 

32. 

(r«-12)(r^  +  20). 

8. 

(c_5)(c-7). 

33. 

(s3-15)(s3  +  8). 

9. 

(,,_6)(r-8). 

34. 

(t^  -  16  W)(f-  4:  w). 

10. 

(r-9)(r-10). 

35. 

(a^-lSy)(x^-^5y), 

11. 

(r  +  2)(r-8). 

36. 

(a-\-lTt')(a-3t'). 

12. 

(m  +  3)(m-10). 

37. 

(x-9a){x  +  15a). 

13. 

(n  +  5)(n- 11), 

38. 

(x-hllb)(x~Sb). 

14. 

{s  +  6)(s- 12), 

39. 

(f  -  13  z)(y' -7  z). 

15. 

(t  +  m-11). 

40. 

(t -i- 15  r)(t- 10  r). 

16. 

(x-2){x-^10). 

41. 

(a'-Ub)(a'-10b) 

17. 

(y  +  ll){y-3). 

42. 

(c-22x)(c  +  20x). 

18. 

{z^l0){z-5). 

43. 

(x  +  19y)(x-5y). 

19. 

(w-4:)(w  +  9). 

44. 

(a -33  b)(a  + 6b). 

20. 

(«-l)(a+8). 

45. 

(c' +  25  d)(c'- 10  d). 

21. 

(s2_6)(s2  +  10). 

46. 

{x-35)(x  +  20). 

22. 

(^_12)(^  +  9). 

47. 

(m-i)(m-5). 

23. 

(xy-10){xy  +  15). 

48. 

(.+!)(. +  1). 

24. 

(r-5s){r+Ss). 

49. 

(2/  +  i)(2/  +  3). 

25. 

(x-2y)(x  +  Sy). 

50. 

(^  +  i)(^  +  i). 

51.    Find  mentally  the  product  of  62  x  68. 

Solution  :  62  x  68  =  (60  +  2)  (60  +  8)  =  3600  +  600  +  16  =  ? 


SPECIAL  PRODUCTS   AND   FACTORING  129 

Find  mentally  the  following  arithmetical  product^: 
62.   22  X  28.  55.    24  x  26.  58.    52  x  54. 

53.   33  X  37.  56.   32  x  38.  59.   23  x  25. 

64.   34x36.  57.    44x46.  60.    33x34. 

Perform  mentally  the  multiplications  in  the  following 
problems : 

61.  Find  the  area  of  a  rectangle  whose  base  is  26  inches 
and  altitude  is  24  inches. 

62.  Find  the  cost  of  12  yards  of  lawn  at  18  ^  per  yard. 

63.  Find  the  cost  of  22  yards  of  scrim  at  25  ^  per  yard. 

64.  Find  the  cost  of  53  bushels  of  corn  at  55  ^  per  bushel. 

65.  Find  the  cost  of  an  11-pound  roast  at  18  ^  per  pound. 

101.   Factoring  Trinomials  of  the  Form  :^  -\-px-\-q. 

A  trinomial  of  the  form  y?-\-i)x  +  q  can  be  factored  if  it  is 
the  product  of  two  binomials  having  a  common  term  (§  100). 

Development.     1.    {x -\- 5)(if  —  3)  =  s? -\-2x—  15. 

In  obtaining  this  product,  the  algebraic  sum  of  +  5  and  —  3  is  taken 
for  the  coefficient  of  oc,  and  the  product  of  +  5  and  —  3  is  taken  for  the 
third  term. 

To  factor  x^  +  2  x  —  15,  it  is  necessary  to  find  two  numbers  whose 
product  is  —  16,  and  whose  algebraic  sum  is  +  2. 

2.   Factor  «2  _|_  7  ^  ^  12. 

Two  numbers  whose  product  is  +  12,  are  -j-  3  and  +  4,  and  their  sum 
is  +  7.     Try  as  the  factors  {x  +  3)  and  (x  -f-  4). 

Check  :  Does  (x  +  3)  (x  +  4)  =  x2  +  7  x  +  12  ?    Yes. 

Rule.  —  To  factor  a  trinomial  of  the  form  x-  -\- px  -\-  q: 

1.  Find  two  numbers  whose  algebraic  product  is  g  and  whose  al- 
gebraic sum  is  p. 

2.  One  factor  is  jt  +  one  number ;  the  other  factor  is  x  4  the 
other  number. 


130  ALGEBRA 

Example  1.     Factor  x^-26x-  192. 

Solution  :  It  is  necessary  to  find  two  numbers  whose  sum  is  —  26  and 
whose  product  is  —  192  ;  the  number  of  greater  absolute  value  must  be 
negative.  If  necessary  write  all  possible  pairs  of  factors  of  —  192,  one  of 
which  is  +  and  the  other  — .     We  have  : 

(4-l)x(-192);  sum  =  -191.         (+4)x(-48);  sum  =-44. 

(4-2)x(-    96);  sum  =  -    94.  (+ 6)  x  (- 32);  sum  =  -  26. 

(+8)x(-    64);  sum  =  -    61. 

.-.  +  6  and  —  32  are  the  numbers  required. 
.-.  a:2  -  26  X  -  192  =  (a:  +  6) (x  -  32). 

EXERCISE  62 
Factor : 

1.  or +  14  a; +  45. 

Solution  :  1.  Find  two  numbers  whose  product  is  45  and  whose  sum 
is  +14. 

2.  Factors  of  45  are,  1  and  45  ;  3  and  15  ;  5  and  9. 

3.  5  +  9  =  14,  hence  the  factors  are  (x  +  5)(a;  +  9). 
Check  :   (x  +  5)  (x  +  9)  =  x^  +  14  x  +  45. 

Note,  This  solution  should  all  be  done  mentally ;  decide  upon  a  pair  of 
factors  of  45  and  immediately  determine  their  sum. 

2.  x'-^-Bx-^-G,  5.    a2  +  lla  +  28. 

3.  a^  +  8x  +  15.  6.    m2  +  9m  +  20. 

4.  ,^  +  10r  +  21.  7.   f  +  9t+lS. 

8.  y_i2p  +  32. 

Solution  :  1.  Since  32  is  positive,  the  second  terms  of  the  factors 
must  have  the  same  sign ;  since  the  sum  of  the  second  terms  is  —  12,  the 
sfecond  terms  are  negative. 

2.    (-8)x(-4)= +32;  and  (-8)  +  (-4)=-12. 

Therefore  the  factors  are  (p  —  8)  (p  —  4). 

Check  :  (j?  -  8)  (j9  -  4)  =  p2  _  i2p  +  32. 

9.    aj2-7a?+12.  12.    iv^  -  10  w -\- 24. 

10.  2/2- 9?/ +  14.  13.    a^- 11  a +  30. 

11.  z'-llz  +  24..  14.   c2-12c  +  35. 


SPECIAL  PRODUCTS  AND  FACTORING  131 

15.  x2_|_6.T_16. 

Solution  :  Since  —  16  is  negative,  the  second  terms  of  the  factors 
must  have  unlike  signs ;  since  the  sum  of  the  second  terms  is  +  6,  the 
term  of  greater  absolute  value  must  be  positive. 

2.  Factors  of  -  16  of  this  sort  are  :  (+16,  -  1)   and  (+8,  —  2). 

3.  (_|_  8)  4-  (  -  2)  =  +  6,  therefore  the  factors  are  (x  +  8)  and  (x  —  2). 
Check:  (x  +  8)(x  -  2)  =  x2  +  6  x  -  16.     (By  §  100.) 

16.  a^  +  3a;-40.  19.   z--\-4:Z-32. 

17.  a:^ -}- 2  a;  -  24.  20.    w^-\-4:W-60. 

18.  y--\-4:y'-21.  21.    a^  +  Sa-Sl. 

22.    ?7i2  -  4  m  —  21. 

SoLDTiON  :  1.  The  factors  of  —  21  must  have  unlike  signs  ;  the  one  of 
greater  absolute  value  is  negative,  since  the  sum  is  —  4. 

2.  Such  factors  of  -  21  are  :  -  21  and  +  1 ;   -  7  and  +  3, 

3.  (  —  7)  +  ( -t-  3)  =  —  4  ;  therefore  the  factors  are  (m  —  7)  and  (m  +  3). 
Check  :  (m  -  7)  (m  +  3)  =  m^  -  4  ?n  -  21.     (See  §  100.) 

23.  x^-2x-  35.  36.  a^-13xw-\-22  w\ 

24.  n^-7n-  18.     "  37.  m^  -  7  m  -  44. 

25.  a^-8a-33.  38.  r^  +  2  r^s  -  48  s^. 

26.  7^  -  4  r  -  45.  39.  s^  -  6  st  -  55  t'\ 

27.  m*  -  11  m2  4-  30.  40.  w^  -  18  tv^  +  72. 

28.  a^  -  15  a  +  54.  41.  /  -f  4  ?/  -  96. 

29.  r^-\-r-2.  42.  a^-70—3a.   (Rearrange  it.) 

30.  X*  +  6  3^-^  8.  43.  6^  -  24  -  10  b. 

31.  c^ -^  15  cd-h  36  d\  44.  c^  +  20  c -f- 84. 

32.  x^  -12xy-h  32  y\  45.  z^  -  36  4-  5  z. 

33.  z^  -{-2z~  63.  46.  a'  +  25  a^  -  150. 

34.  c^d'^  -cd-6,  47.  -  19  m^  +  m^  -f  84. 

35.  2/-  +  10  7/2  +  9  z\  48.  a^^  -h  50  -  27  x". 


132  ALGEBRA 

49.  t^-st-20s\  '      55.  a^ft^  -  16  a6c  + 28  cl 

50.  a"  -\-l  ah  -m  h\  56.  x^  -  21  xyz  +  108  yh\ 

51.  ^2  -  20  aa;  +  99  a''*.  57.  a" -- 21  ah'' +  110  h\ 

52.  r^  4-  rs  -  72  si  58.  c^d^  -  10  cd  -  96. 

53.  x^ +  ^Oy'-l^xy\  59.  aV  -  26  ac  + 160. 

54.  a2  +  16a  +  15.  60.  r^  -  7  rs  -  120  s^. 

Find  all  of  the  prime  factors : 

61.  3  ah^  _  15  a&  +  18  a. 

Solution  :     ^  ah'^  -  lb  ah  +  1%  a  -  ^  a  {h'^  -  bh  ■\-  Q) 

62.  5f  +  S5t-i-  60.  66.    ahn^  -  7  abn  -}- 6  ah. 

63.  mr2  +  2  mr  -  15  m.  67.   11  a^  -  11  a;  -  66. 
6^.   Sca^  +  6cx-9  c.                   68.    8  aj^  +  48  a;  -  56. 

65.    7  yh  ^  21  yz  -  126  z.  69.   m^ri^  -\-2m^n-  63  ml 

CASE  V 

Ti/P^  i^brm  .•  (ax  +  6)  (cjf  +  d). 
102.    The  Product  of  Two  Binomials  of  the  Form  (ax  +  b). 

Development.  1.  Find  by  actual  multiplication  the  fol- 
lowing products,  and  write  the  results  as  in  part  (a) : 

(a)  (2  a!  +  3)  (3  a;  4- 1)  =  6  a^  +  11  a:  +  3. 

(h)  (3m  +  2)(2m  +  3)=?         (c)  (2y -j-A)(3y -2)=? 

2.  Examine  carefully  the  results  in  1,  then  try  to  get  the 
first  and  third  terms  of  the  following  products  mentally  ;  pos- 
sibly you  can  get  the  second  term  also.  Check  by  actual 
multiplication. 

(a)  (2  r  4-  4)(3  r  +  1).  (c)  (3  s  +  5)(2s  +  1). 

(h)   (2r  +  5)(r  +  2).      .  (d)  (4.x  +  2y)(2x  +  ^y).- 


SPECIAL  PRODUCTS   AND   FACTORING  138 

3.    Below  is  given  the  product  of  (5  x  —  4  y)  and  (2  x  -|-  3  y). 
5x  —  4:y 
2x  -\-3y 
10  ar*-    Sxy 

-\-15xy-12f 
10ar^+    lxy-12f 


...  (5a;_4^)  {2x  -h  ^y)  =  10x^  +  {-^  +  15)xy  -12^. 

Rule.  —  1.  The  first  term  of  the  product  is  the  product  of  the  first 
terms  of  the  binomials,     {bx   2x  =  l0x^.) 

2.  The  third  term  of  the  product  is  the  product  of  the  second 
terms  of  the  binomials,     {—^y)    ( +  3 1/)  =  -  12 y-. 

3.  The  second  term  of  the  product  is  the  algebraic  sum  of  the 
"  cross  products."  (Notice  the  position  of  these  cross  products  on 
the  left  of  the  equation  :    —^y-2x  and  b  xZy). 


Example  1.    Find  the  product  (5  r  —  6  s)  (2  r+  3  s). 

Solution  :  In  all  of  these  examples,  the  only  difficulty  is  that  of  find- 
ing the  "  middle  term." 

1.  5r-38  =  15rs,  -6s.2r  =-  12rs,  and  (15rg)  +  (-12rs)  =  -|- 3  rs. 

2.  .-.  (5  r  -  6  s)  (2  r  +  3  s)  =  10  r2  +  3  rs  -  18  s\ 

Example  2.   Find  the  product  (9  a;  -f  4  2/)  (3  a;  —  6  ,y). 

Solution  :  9  •  -  6  =-  54,  4  •  3  =  12,  and  (-54)  +  (12)  =-42. 

.-.  (9x  -f  4  y)  (3  X  -  6  y)  =  27  a;2  _  42  xy  -  24  y^. 

In  this  example,  the  coefficient  of  the  middle  term  was  found  first  and 
then  xy  was  affixed.  In  practice  all  of  this  should  be  done  mentally  as  in 
the  following  example. 

Example  3.   Find  the  product  (7  m  —  4n)(8mH-5n). 
Solution  :  (7  ?n  -  4  n)  (8  w  -f  5  w)  =  56  m^  +  3  mn  -  20  n^. 
[Middle  terra  :  7  •  6  =  35,  -  4  •  8  =  -  32,  35  +(-  32)  =3.] 
Pupils  should  try  to  acquire  such  skill  that  they  can  find  the  correct 
products  mentally.     This  is  the  manner  in  which  experts  do  it. 


134  ALGEBRA 

EXERCISE  63 

1.  (2x  +  2)<ix  +  3).  27.  (9c^-2)(6c^  +  l). 

2.  (2x^S)(x-{-2).  '28.  (10n^-^3)(15n'-4:). 

3.  (2  a;  4- 1)0^ +  3).  29.  (14  ar^- 5) (2 a^  +  1). 

4.  (3x+l)(2a;  +  3).  30.  (16^-9)(3f^ +-2). 

5.  (2iB  +  3)(3£c  +  2).  31.  (14rs-9)(5rs  +  3). 

6.  (3a-2)(2a-l).  32.  (2  a-Sb){4.a  +  5b). 

7.  (3a-4)(a-2).  33.  (6m  +  5s)(7m-4s). 

8.  (3a-4)(a-3).  34.  (St- 3x)(9t-x). 

9.  (3a-5)(2a-3).  35.  (.t- Si/) (a; +  5 2/). 

10.  (4a-5)(2a-5).  36.  (2  a; -3  2/)  (2  a;  +  32/). 

11.  (3r  +  5)(2r-l).  37.  (2  m- 5n)(2m- 5n). 

12.  (4r  +  7)(2r-3).  33.  (3t-\-A7i)(3i-\-4.n). 

13.  (5r  +  4)(4r-2).  39.  (9  a6  -  4c)(3a6  +  5  c). 

14.  (7r  +  6)(3r-2).  40.  (6a;2/- '^^)(5a'2/  + 62;). 

15.  (8rH-9)(4r-3).  41.  (7?'2  +  8s)(8?^-9s). 

16.  (6  s -5)  (3  s +  2).  42.  (10a:3_ii2/3)(llaj3_^122/^). 

17.  (10s-7)(4s+l).  43.  (9m?i2  4-4)(6m7i2-3). 

18.  .(lls-12)(5s4-4).  44.  (2  +  9»)(3  +  2a;). 

19.  (9s-12)(4s  +  5).  45.  (5-7 f-)(6-{-9f). 

20.  (15s-20)(3s  +  2).  46.  (8  +  3;2^)(10- 42^). 

21.  (6ab  +  2)(3ab-5).  47.  (7 -6xy){5  +  6xy). 

22.  (7w?i-4)(6mn-3).  48.  (9  -  11  xy^)(4.  -  xy^). 

23.  (8r2-3)(9r2  +  4).  49.  (10  a'b -6  c){9  a'b- 2  c). 

24.  (13a^  +  7)(5a^-3).  50.  (12  a^^-f- 7  2/) (5  ar'- 4  2/). 

25.  (6p'-7)(4.p'-\-3).  51.  (9^2-5a^)(7«2  +  5a;-> 

26.  (llm2-4)(5m2  +  6).  62.  (Sm^- 9  M)(4m2  4-5  n>mq 


SPECIAL  PRODUCTS   AND   FACTORING  135 

53.  (7ar^  +  3/)(5ar^-2  2r).  57.  (lom-2n)(6m -h  5n). 

54.  (llc2-5cZ)(4c2-5d).  58.  (6m  +  |)(4m  +  i). 

55.  (15p'-^2q)(92^-2q).  59.  (^  x-^y){9  x- iy). 

56.  C20a-7b)(Sa-hSb}.  60.  (12a  + f  6)(10  a-|  6). 

103.    Factoring  Trinomials  of  the  Form  mjr  +  nx  +  />.      The 

product  of  two  binomials  like  (2  a; -|-  3)  and  (3x+5)  is  a  bi- 
nomial of  the  form  mx^  +  nx  -|-  p. 

This  means  that  there  is  a  term  containing  the  second  power  of  x,  one 
containing  the  first  power  (as  a  rule),  and  one  free  from  x. 

The  following  discussion  shows  how  to  factor  trinomials  of 
the  form  m3i?-\-nx-^p,  when  they  are  factorable. 

Development  1.     Find  the  products : 

(a)  (2a;-f  6)(3a;  +  2).  (6)  (3  a! -5) (4  a: +  7). 

2.  How  do  you  obtain  the  "  middle  term  "  of  the  product  ? 

3.  Factor  12  a^  +  23  a;  +  5. 

Solution  :  1,  The  first  terms  of  the  binomials  might  be  2  a-  and  6  x,  for 
their  product  is  12  x'^.     Place  them  in  parentheses  thus  :    (2x      )  (6  z      ). 

2.  The  second  ternis  of  the  binomials  are  both  positive  since  5  and 
23  X  are  positive.  Place  the  factors  -|-  5  and  +  1  in  the  parentheses 
and  note  the  middle  term  which  results. 

(a)   (2  a;  +  6)(6  a;  +  1)  ;  middle  term,  +  32  a;.     Incorrect. 
(6)  (2  X  +  1)(6  X  +  6)  ;  middle  term,  +  16  x.     Incorrect. 

3.  Step  2  shows  that  the  factors  2  x  and  6  x  for  12  a;^  are  incorrect. 
Try  3  X  and  4  X  for  12  x2,  thus :  (3  X      )(4x      ). 

(a)  (3x -f  l)(4x  +  6)  ;  middle  term,  +  19x.     Incorrect. 
(6)  (3  X  +  6)(4  X  +  1)  ;  middle  term,  +  23  x.     Correct. 
Check  :  (3  x  +  5)(4  x  +  1)  =  12  x^  +  23  x  +  5. 

NoTK.  This  may  seem  a  long  process  at  first,  but  practice  soon  develops 
such  skill  that  most  of  the  trial  of  factors  can  be  done  mentally. 


186  ALGEBRA 

EXERCISE  64 
Factor : 

1.  Sx^-i-5x  +  2.  5.  6a;2  +  7a;  +  2. 

2.  Sa^^7a  +  2.  6.  12f-^13ti-S. 

3.  5m2  +  7?ii  +  2.  7.  7a2  +  l0a  +  3. 

4.  7/  +  9:f/  +  2.  8.  6^-  +  13«  +  6. 
9.  12i^2~17w  +  6. 

Solution  :  1.  Since  6  is  positive,  its  factors  must  have  like  signs ; 
and,  since  —17  is  negative,  the  cross  products  must  both  be  negative. 
Therefore  the  factors  of  6  must  both  be  negative. 

2.  To  get  12  W52,  use  {2w       )(()W      ). 
To  get  +  6,  try  -  2  and  -  3. 

(a)  (2  10  —  3)  (6 10  —  2) ;  middle  term,  —  22  w.     Incorrect. 
(6) '  (2io  —  2)(6w  —  Z);  middle  term,  —  18  w.    Incorrect. 

3.  To  get  12  io2,  use  (3  wj      ) (4  w      ). 
To  get  +  6,  try  —  2  and  -  3. 

(a)   (3  tc  -  2) (4  «?  -  3)  ;  middle  term,  —  17  to.     Correct. 
Check:  Does  (3io  -  2)(4  wj  -  3)=  12  m?2  _  17  ^^  +  G  ?     Yes. 

Note.  In  step  2,  the  factors  3  and  2  were  used  in  both  ways,  as  in  (a)  and 
(&) ;  usually  this  is  a  wise  plan,  although  in  this  case  an  explanation  could  be 
given  to  show  that  it  was  unnecessary. 

10.  2w^  —  llw-\'5.  14.  2w;2_9it'  +  4. 

11.  2w'~-7iv-{-3.  15.  6m2-ll7/i  +  3. 

12.  Sf/-19b-^6.  16.  8«2_22a  +  15. 

13.  6m^-7m  +  2.  17.  5c'' -23c -\- 12. 

1 8.   Factor  15  x-  4- 14  a?  -  8. 

Solution:  1.    The  factors  of  —8  must  have  unlike  signs.     Arrange 
the  signs  so  that  the  cross  product  of  greater  absolute  value  is  positive. 
2.   For  15a:2,  try  (3x       )(5x      ).     For  8,  try  2  and  4. 
(a)  (3  X  —  2)  (5  a-  +  4)  ;  middle  term,  +2x.     Incorrect. 
The  sign  of  4  is  made  +,  because  4  •  3  is  greater  than  2  •  5. 
(6)    (3  X  +  4) (5  X  -  2)  ;  middle  term,  +  14  x.     Correct. 
The  sign  of  4  is  made  +,  because  4  •  5  is  greater  than  3  •  2. 
Check:  Does  (3x  +  4)(5x  -  2)=  15x2  +  14x  -  8  ?     Yes. 


SPECIAL  PRODUCTS  AND  FACTORING  137 

19.  Factor  24 m^ -m-  10. 

SoLDTioN  :  1.  The  factors  of  —10  must  have  unlike  signs.  The  cross 
product  of  greater  absolute  value  must  be  negative. 

2.  For  24  m"^,  try  (0  7rt       )(4  m       ).     For  10,  try  5  and  2. 
(a)  (6  m  4-  2) (4  ?>i  —  6) ;  middle  term,  —  22  m.     Incorrect. 
(6)   (6  m  —  6)  (4  »n  +  2)  ;  middle  term,  —  8  m.     Incorrect. 

3.  For  24  m^,  try  (3  m      ) (8  w      ). 

(a)  (3  m  —  6)  (8  m  4- 2) ;  middle  term,  —34  m.    Incorrect. 
(6)   (3  m  —  2)  (8  m  +  5)  ;  middle  term,  —  m.     Correct. 
Check  :  Does  (3  m-  2)  (8  ?»  +  5)  =  24  m^  _  m  -  10  ?    Yes. 

Note.  In  this  last  step,  for  example,  after  placing  2  and  5  in  the  parenthe- 
ses, we  see  that  2  x  8  is  greater  than  3x5;  therefore  make  2  negative  and  5, 
positive. 

20.  2a^-3a;-5.  37.  14  mV  -  31  mn  - 10. 

21.  3»i*  +  4m-7.  38.  4:i^s^ +  irs-15. 

22.  5t^-2t'-7.  39.  12 ic^ 4- 17 a;  +  6'. 

23.  7?'2-|-4r-ll.  40.  24m«-18m-15. 

24.  6s2-7s-5.  41.  12 ^^ _^  13 ^a; - 35 a:^ 

25.  1282-f5s-3.  42.  2w'-3wr-20r'. 

26.  2x2  +  a;-15.  43.  15y' ■\- 19 yz-\- 6 z'. 

27.  9r2-6r-8.  44.  lOn^ -^9nt-9t\ 

28.  15c2-4c-3.  45.  9/r +  3  ftc-SGc^. 

29.  21A'  +  2A-S.  46.  15a2_26a6 -21  61 

30.  5/  +  16y  +  3.  47.  7ar*-26ir.y-8?/. 

31.  18«2_3^_10.  48.  15a2-|-29a?>4-12  62. 

32.  4:x'-24:X  +  S5.  49.  21  a;^  _  29  xt/ - 10  y*. 

33.  10ar'-13a;-30.  50.  G  r"  -  2rj  rs -\- 25  s". 

34.  322_|.22z4-7.  51.  4c2-8cd-21d2. 

35.  18m2-f-17m-15.  52.  9d*-6d2~35. 

36.  6x2 +  31  a; +  35.  63.  8i)«  +  18p»-35. 


188  ALGEBRA 

54.  10  m* -\- 19  m^n  —  15  nK  57.  35  df -\- at  —  12  a. 

55.  Uv'-\-29v-15.  58.  24.x^7^-7xy7^-6y^r^. 

56.  15k'^x  —  16kx  —  15x.  59.  55  m V+2  mnar*— 21/1 V. 
(Remove  the  monomial  first.)  60.  24  C^d^  -\-2cd^  —  15  d^. 

EXERCISE  65 

Perform  the  various  steps  of  the  following  indicated  multi- 
plications mentally : 

1.  3a(2a  +  5)(a-4).. 

Solution  :  3  a(2a  +  5)  (a  -  4)  =  3  a(2  a^  -  3  a  -  20) 

=  6  a^  -  9  a2  -  60  o. 

2.  2m(3m-l)(2m  +  6).         5.   5(2r-5s)l 

3.  4.ab(5x-2){5x-{-2).  6.    -3(6  a- 5)(2  a  +  3). 

4.  7(3cH-5d)(3c-4d).  7.    -2(7  r-4  s)(2r4-3  s) 

Solve  the  following  equations,  performing  mentally  all  of 
the  steps  of  the  solution  : 

8.  2(3x-2)(x-^4:)-(6x-^5)(x-S)  =  65. 
Solution  :  1.    2(3  x  -  2)  (x  +  4)  -  (6  x  +  5)  (x  -  3)  =  65. 

2.  2(3x2  +  10x-8)  -  (6x2-13x-15)  =65. 

3.  Complete  this  example  by  removing  parentheses,  combining  terms, 
etc. 

9.  (5  a;  -  4) (3  a;  -h  4)  -  3  (5  aj  +  6) (a;  -  7)  =  10  (9  a;  +  15). 
10.    (4  r  -  5)(3  r  +  7)  -  2 (r  + 1)(6  r  -  7)  =  3  (2  r  +  3). 

Find  all  of  the  prime  factors  in  the  following ;  remember 
to  remove  the  monomial  factor  first : 

11.  24  m^a  H- 18  mna  — 15  A. 

12.  lSbc'-2bx'. 

13.  12xHj-Sx'y^-4.x^f. 

14.  15  c^d  -  We'd- 25  cd., 

15.  IS  Ic'r"-- 60  Mr' -{- 501^7^. 

16.  4.S  cd'  + 120  cde  + 75  ceK 


SPECIAL   PRODUCTS  AND   FACTORING  139 

17.  39  m-nar  -f-  20  mhixy  -  4  mhiy\ 

18.  3x*-17x'-mx'. 

19.  ■ixy-^2x^f-7a^y\ 

20.  4:5  r*s  -  SO  r^s'. 

CASE  VI 
Type  Form :  a^-b^  =  (a-  b)(a-  +  a6  +  ft^). 
104.   Factoring  the  Difference  of  Two  Cubes. 
Development.     1.   Divide  a^  —  27  by  a;  —  3 ;  by  » -}-  3. 

2.  Divide  x^ —  y^hj  x  —  y;  hy  x-\-y. 

3.  Write  the  cube  of  a;  the  cube  of  6;  the  difference  of 
these  cubes. 

4.  Examine  the  results  in  1  and  2 ;  by  what  do  you  think 
you  can  divide  a^—  b^  and  get  an  exact  quotient? 

5.  Prove  by  division  the  following  fact : 

Hence,  «» -  6«  =  (a  -  b)(a'  -\-ab-\-  b^. 

Rule.  —  1.  The  difference  of  the  cubes  of  two  numbers  may  be 
divided  exactly  by  the  difference  of  the  numbers. 

2.  The  quotient  is  the  square  of  the  first  number,  plus  the  product 
of  the  two  numbers,  plus  the  square  of  the  second  number. 

Example  1.     Find  a  divisor  of  8  a^  —  27 ;  find  the  quotient. 
Solution  :  1.   8  x8  -  27  =  (2  x)8  _  38. 

2.  .'.  It  is  the  difference  of  the  cubes  of  2  a:  and  3,  and  can  be  divided 
exactly  by  (2x  -3). 

3.  .-.  The  quotient  is  (2  xy  +  (2  x)(3)  +  (3)2,  or 

4  x2  +  6  :k  +  9. 
Check  by  substitution  or  by  multiplication. 

Example  2.     Factor  8  a^  —  64  i/^. 

Solution  :  1.   8  x^  -  64  2/«  =  (2  x^)^  -  (4  yY 

2.  =(2x2-4y){(2x2)2+(2x2)(4y)  +  (4  2/)2} 

3.  =(2x2-4«/){4x4  +  8x2?/  +  16  2/2}. 

NoTK.  The  middle  term  is  not  twice  the  product  of  the  first  and  second 
numbers ;  try  not  to  confuse  this  type  with  the  one  in  §  96. 


140 


ALGEBRA 


EXERCISE  66 


Factor : 

1.  a'-S. 

2.  63_(34. 

3.  c^  — 125. 

4.  c'-SdK 

5.  m^  —  64  n^. 

6.  r«-8. 

7.  s«-125. 

8.  a^2_g4^s^ 

9.  Sb^-a^ 
10.  64m«-l. 


11.  8a;«-125?/3. 

12.  3ar^-24?/«. 
(Remove  monomial  first.) 

13.  AOa'-Bb^ 

14.  24aj<'-81?/l 

15.  4.a  —  500ax^ 

16.  320m3-5w«. 

17.  a^-i- 

18.  /-^. 


19.    z^- 


125 


What  is  the  quotient  of 

20.  (a^-a3)^(a;-a)? 

21.  (7^-x')-^(r-x^)? 


24.  (S-x^)-^(2-x')? 

25.  (mi^-27)-^K-3)? 

22.  (27a.-«//-l)--(3ir2/-l)?   26.  (27 w'-6iv^)-r-(3w'~Av)? 

23.  (64a«-6«)-(4a2-?>2)?   27.  (1  -  125  m«)  -  (1  -  5  m^)? 


CASE  VII 
Type  Form :     (a^  +  b')  =  (a  +  &)  (a'  -  fl&  +  &")• 
105.   Factoring  the  Sum  of  Two  Cubes. 

Development.     1.    Divide  (a^  +  27)  by  (x  +  3) ;  hj  x  —  3. 

2.  Write  the  cube  of  m;    of  7i;    the  sum  of  these  results. 
Write  the  sum  of  m  and  n. 

3.  Divide  the  sum  of  the  cubes  of  m  and  n,  by  the  sum  of 
m  and  n;  also  by  their  difference. 

4.  Prove  by  division  the  following  fact : 

(a"  +  b^)  -^  (a  -{-b)  =  a^  -  ab  +  b\ 
Hence  (a^  +  6»)  =  (a  +  ?>)(a-  -  tt6  +  Z>')- 


SPECIAL   PRODUCTS   AND   FACTORING  141 

Rule  1.  —  The  sum  of  the  cubes  of  two  numbers  may  be  divided 
exactly  by  the  sum  of  the  two  numbers. 

2.  The  quotient  is  the  square  of  the  first  number,  minus  the  prod- 
uct of  the  two  numbers,  plus  the  square  of  the  second  number. 

Example  1.  Find  an  exact  divisor  of  x*^  -}-  8 ;  find  the 
quotient. 

SoLL'TioN :  a*  -f  8  =  (x:^y  +  (2)8. 

Since  it  is  the  sum  of  the  cubes  of  two  numbers,  it  may  be  divided  ex- 
actly by  the  sum  of  the  two  immbers  (jc^  -j-  2).     (Rule  1.) 

The  quotient  is  {x^y  -  (a;2)  .  (2)  +  (2)2  or  x*  -  2  x  +  4.     (Rule  2.) 

Note.    The  middle  term  is  not  twice  the  product  of  the  two  nambers. 

Example  2.     Find  the  prime  factors  of  3  mx^  +  81  mi^. 

Solution  :  3  wx^  +  81  ?«?/«  =  3  m{x^  +  27  f). 

'  =3m[(x)3+(3y2)3-| 
=  3  m[x  +  8  2/2]  [;r2  -  (3  y^)  •  x  +  (3  y2)a] 
=  3  w(x  4-  8  y2)(:i--i  -  3  xy'^  +  9  y^). 

EXERCISE  67 

Change  the  sign  in  each  example  of  Exercise  66  from  minus 
to  plus.     This  will  give  27  examples. 

106.  Summary.  In  this  chapter,  seven  special  forms  of 
algebraic  expressions  have  been  considered.  These  type  forms 
are  collected  here  for  reference : 

I.    a{x  +  y  +  z-\-.  .  .)   =ax+ay-\-az  +  .  .  .  . 
II.    {a±bf  =a2±2a&+&2^ 

III.  {a-\-h){a-h)  =d'-tr. 

IV.  (x  +  a){x  +  b)  =x'-^(a-{-b)x-^ab. 

V.    {ax-^  b)  {ex  -\-d)        =  acxr  +  {ad -\-bc)x-\-  bd. 

VI.    (fl-6)(a-4-a&  +  6')  =  ff'-fr'- 
VII.    (a  +  b){(^ -ab^b")  =a?-\-b\ 
Familiarity  with  these  forms  makes  it  possible : 
(a)  to  perform  many  multiplications  in  algebra  and  arith- 
metic mentally  ; 


142  ALGEBRA 

(6)  to  perform  many  divisions  mentally  ; 
(c)  to  factor  many  algebraic  expressions. 
These  are  the  more  important  forms.     Others  are  discussed 
in  a  later  chapter.     (XVI.) 

Rule.  —  To  perform  an  indicated  multiplication  of  two  or  more 
factors  by  means  of  the  type  forms : 

1.  First  find  the  product  of  all  binomial  and  poljmomial  factors. 

2.  Then  multiply  by  the  monomial  factor. 

Example.     Find  the  product  Sx(2  x-\-y)(2x  —  y)(4:X^ -\-y^. 
Solution  :  S  x(2x -h  y)(2x  -  y)(4x -{-  y^) 

=  3  X  .  (16  a;4  _  y^) 
=  48  x^  -Sxy*. 

Rule.  —  To  find  the  prime  factors  of  a  given  expression : 

1.  First  remove  any  monomial  factor  of  the  expression. 

2.  Factor  the  resulting  polynomial  factor  or  factors  by  the  proper 
methods,  until  all  of  the  prime  factors  have  been  found. 

Example.     Factor  3  ax^  —  21  aoif  —  24  a. 
Solution  :  3  ax^  —  21  ax^.  —  24:  a 

=  3  a(x^  -  7  x3  -  8) 

=  3a(x3-8)(x3  +  l) 

=  3a(x-  2)(x2  +  2  X  +  4)(x  -f  l)(x2  -  x  +  1). 

EXERCISE  68 
MISCELLANEOUS  EXAMPLES 

Expand  the  following  expressions  : 

1.  (3x-\-2y)(4:X-3y).  6.  (a +  6)(a- 6)(a2  + 6=^. 

2.  (3^x-6y)(j\x-^6y).  7.  (9x-2y)(x-\-y). 

3.  (a'  +  a^  ■\-  l)(a2  - 1).  8.  3  x(a  -b)(a  +  b). 

4.  (2  a -by.  9.  (5a-4b^(5a-6b'). 

5.  (7  a^  +  b-){S  a2  -  8  b^.  .  10.  3  m(m*  -  4  mhi^  +  2  7i% 


SPECIAL   PRODUCTS   AND   FACTORING  141 

11.  7/i(n«-Hl)(n3-l).  21.  3y(2  -  y)(5 +3,v). 

12.  3a(a-|-l)(a-l).  22.   7(3  m  + l)(m- 6). 

13.  {a?  -  12  y){a^  -f  9  y).  23.  {x^  +f){x'-  y"). 

14.  {x-\-y){x'-xy+y-){x^-f).      24.   (a^y  -  4)(iC2/+ 16). 

15.  (a;-f?/)^  25.  2{x-4.yy. 

16.  (m  -  \){m  -  \).  26.  (a;  +  V){x  -  l)(ar^  -  3). 

17.  2<3a;-hl)(a;-4).  27.   (a^ +i)(a^- i  a;  +  J). 

18.  (B'-AACXS^-eAC).       28.  (Ia^-f)(iar»4-J). 

19.  a(a^  +  3)(a3-3).  29.  (2?i-^)2. 

20.  5(a-6)(2a+3  6).  30.  (.^  -  f)(a; +  f). 

Find  the  prime  factors  of  the  following : 

31.  5 r^- 40a; +  80.  47.  125a^-Sy^. 

32.  aa^  +  6aa:^  +  9a.  48.  Sa^-Sa-S. 

33.  a;22,2  _  5  a^a^y  +  6  a^ftl  49.   169  a^^  78  a&  + 9  6^. 

34.  3x^-60  a^- 288  y.  50.  30  m^  -  47  m  -  5. 

35.  c2  4- 44  c -45.  51.  24:  x^m  +  SI  fm. 

36.  c3-64d^.  52.  a«-f  14a«  +  49. 

37.  3  a:^  ^  24  f.  53.  ar^^/  -  2.16  y*, 

38.  m*+23w='n  +  132?il  54.  3^^  +  33  A; +72. 

39.  a^-18a2  +  77.  55.  9x^-4:Xy-lSf. 

40.  x^-3a^y-10Sy\  56.  33^-3/. 

41.  4  ar^- 28  a;?/ +  49  2/1  57.  a* -256. 

42.  169mV-raV.  58.  3  a^- 108  a- 

43.  a2  +  17a-38.  59.  c'cP +  9  cd-52. 

44.  3c2  +  132c-135.  60.  5x^^-5  7/^. 

45.  x^-y\  61.  25r2  +  60rs  +  36s*. 

46.  2d2_26d-136.  62.  ar^  +  Ja;  +  ^. 


[4: 

ALGEBRA 

63. 

196f-^\%a\ 

67.  xh^-20xz-69. 

64. 

-  25  a;  +  «2  -f- 100. 

68.  18  -  19  c+c2. 

65. 

300  m*  -  243  ar^. 

69.  b  -  K)  +  15  b\ 

66. 

y  + 12  y  _  108. 

70.   —  50a  +  a2  +  49. 

107.    The  skill  acquired  in  this  chapter  in  performing  some 
multiplications  mentally  is  of  use  in  solving  certain  equations. 

Example.     Solve  the  equation  : 

6(4.-xy-5{2x-\-7){x-2)  =  22-{2x  +  Sy. 
Solution  :  Perform  the  operations  mentally. 

1.  6(4~a:)2_5(2a;  +  7)(x-2)=22  _(2  x  +  3)2. 

2.  .  •.  6(16  -  8  X  +  x2) -  5(2  x-^  +  3  a;  -  14)  r^  22  -  (4  ic2  +  12  a;  +  9). 

3.  .-.  96  -  48  X  +  6  x2  -  10  x^  -  15  x  +  70  =  22  -  4  x^  -  12  x  -  9. 

4.  .-.  166-63x-4x2=13- 12x-4x2. 

5.  .-.  -  63  x  + 12x3=  13 -166. 

6.  .-. -51x=- 153. 

7.  .-.  x  =  3. 

1  13  1  81 

Check  :  Does  6(|^-=^)2_  6(6,.f-^ (lU^  z=  22  -  66-r^)2  ? 

Does  6  -  65  =  22  -  81  ? 

Does  -  59  =  -  59  ?    Yes. 

EXERCISE  69 

Solve  the  following  examples  performing  all  of  the  work 
mentally : 

1.  (5ic  +  7)(3x-8)  =  (5a;  +  4)(3a;-5). 

2.  (4m-7)«=(2m-5)(8m  +  3)-2. 

3.  (5-3  r)(3  +  4  7-) - (7  +  3  ?-)(l  -  4  r)=  -  28. 

4.  (l_3^)V-(p  +  5)2=40>+l)(2i7-3). 

5.  2t{t-\-7)-(t-5y  =  (t  +  3){t-ll). 

6.  (3a;+5)(2a;-3)-6(a;-2)(ic  +  13)=ll. 

7.  (7a-2)(3a  +  5)-(4a  +  3)2  =  5a(a  +  2)+l. 


SPECIAL  PRODUCTS  AND   FACTORING  146 

8.  (5  n  -  6)  (5  71  +  6)  -  5(2  n  -  3)-  =  5(n  -f  12)  (n  -  1). 

9.  3(4a_5)2_(6a-  1)- =  7 +  (2a  -  9)(6  a- 11). 

10.  (2x-7y-(ox-2y+3x{7x-{-5)  =  -i. 

11.  Find  two  numbers  whose  difference  is  6,  and  the  differ- 
ence of  whose  squares  is  120. 

12.  Find  two  numbers  whose  sum  is  13,  and  the  difference 
of  whose  squares  is  65. 

13.  Divide  the  number  20  into  2  parts  such  that  the  square 
of  one  exceeds  the  square  of  the  otlier  by  40. 

14.  Find  four  consecutive  numbers  such  that  the  product  of 
the  first  and  third  shall  be  less  than  the  product  of  the  second 
and  fourth  by  9. 

15.  Find  two  consecutive  numbers  such  that  the  difference 
of  their  squares,  plus  5  times  the  greater  number,  exceeds  4 
times  the  less  number  by  27. 

16.  Find  two  consecutive  numbers  such  that  the  sum  of 
their  squares  exceeds  twice  the  square  of  the  smaller  by  261. 

17.  One  man  travels  a  certain  distance  in  as  many  hours  as 
he  travels  miles  per  hour ;  another  man  travels  the  same  dis- 
tance in  two  hours  less  time  by  going  three  miles  per  hour 
faster.     What  was  the  rate  and  time  of  the  first  man  ? 

EXERCISE  70 
Problems  about  Area 

1.  Express  the  area  of  each  of  the  following  figures,  assum- 
ing for  each  an  altitude  of  20  inches  and  a  base  of  30  inches : 

(a)  A  rectangle  (§  17).  (b)  A  triangle  (§  17). 

(c)  A  parallelogram  (§17). 

2.  Express  the  same  areas  if  the  altitude  is  2  a;  and  the 
base  (x—  6). 


146  ALGEBRA 

3.  Express  the  same  areas  if  the  base  is  y  and  the  altitude 
exceeds  the  base  by  4. 

4.  The  side  of  a  square  is  s  inches. 

(a)  Express  the  area  of  the  square. 

(b)  Express  the  dimensions  of  a  rectangle  if  its  base  is  4 
inches  more  and  if  its  altitude  is  3  inches  less  than  the  side  of 
the  square. 

,    (c)  Express  the  area  of  the  rectangle. 

{d)  Form  an  equation  expressing  the  fact  that  the  area  of 
the  rectangle  exceeds  the  area  of  the  square  by  50  square 
inches. 

5.  The  base  of  a  rectangle  exceeds  twice  its  altitude  by 
5  inches ;  the  base  of  a  triangle  exceeds  the  base  of  the  rec- 
tangle by  4  inches,  and  its  altitude  exceeds  the  altitude  of  the 
rectangle  by  3  inches.  Let  a  represent  the  altitude  of  the 
rectangle. 

(a)  Express  the  dimensions  and  area  of  the  rectangle. 
(6)  Express  the  dimensions  and  area  of  the  triangle. 

(c)  Form  an  equation  expressing  the  fact  that  the  area  of 
the  rectangle  exceeds  the  area  of  the  triangle  by  25  square 
inches. 

Equations 

6.  The  base  of  -a  certain  rectangle  exceeds  its  altitude  by 
8  inches.  If  the  base  and  altitude  are  both  decreased  by  2 
inches,  the  old  area  exceeds  the  new  by  36  square  inches. 
Find  the  dimensions  of  the  rectangle. 

7.  The  base  of  a  rectangle  is  9  feet  more  and  the  altitude 
is  8  feet  less  than  the  side  of  a  square.  The  area  of  the  rec- 
tangle exceeds  the  area  of  the  square  by  15  square  feet.  Find 
the  dimensions  of  the  rectangle. 

8.  A  man  planned  a  house  whose  length  exceeded  its 
width  by  10  feet.  He  found  that  it  would  be  too  expensive  to 
build   the   house  as  planned,  so  he  decided  to  decrease  both 


SPECIAL   PRODUCTS   AND   FACTORING  147 

dimensions  5  feet.  He  found  that  this  made  a  difference  of 
425  square  feet  in  the  area  covered  by  the  house.  What  were 
the  original  and  the  new  dimensions  ? 

9.  The  main  shaft  of  Washington's  Monument  is  square 
at  the  bottom  and  top.  The  side  of  the  lower  square  exceeds 
the  side  of  the  upper  square  by  21  feet.  The  area  of  the  lower 
square  exceeds  the  area  of  the  upper  square  by  1869  square 
feet.     Find  the  dimensions  of  the  two  squares. 

10.  A  man  planned  to  set  out  an  apple  orchard  with  two 
more  trees  to  each  row  than  he  had  rows,  but  found  that  that 
plan  left  one  tree  over.  He  found  that  if  he  decreased  the 
number  of  rows  by  3,  and  increased  the  number  of  trees  per 
row  by  5,  he  used  all  of  his  trees.     How  many  trees  had  he  ? 

QUADRATIC  EQUATIONS  SOLVED  BY  FACTORING 

108.  Not  all  equations  are  simple  or  first  degree  equations. 
(§77-) 

Example.  Find  the  number  whose  square  exceeds  the  num- 
ber itself  by  6. 

Solution  :  1.  Let  n  =  the  number. 

2.  Then  n^  =  the  square  of  the  number, 

3.  and  .-.  n^  =  n-\-6. 

4.  .-.  n2  -  n  -  6  =  0. 

5.  Factoring  :    (n  -  3) (n  +  2)  =  0, 

6.  If  (w  —  3)  =  0,  then  0  •  (n  +  2)  would  also  equal  zero. 

n  -  3  =  0,    if  n  =  3. 

7.  If  (n  +  2)  =  0,  then  (n  —  3)  •  0  would  also  equal  zero. 

n  4-  2  =  0,    if  n  =  —  2. 

8.  These  values  of  n,  4-3  and  —  2,  should  satisfy  equation  4. 
Check  :  If  n  =  3,     does  (3)2  -3-6  =  9-3-6  =  0?    Yes. 

If  71  =-2,  does  (-2)2-  (-2) -6  =  4  +  2-6  =  0?    Yes. 

9.  Moreover,  both  of  these  numbers  satisfy  the  conditions  of  the 
problem  :  32  is  9 ;        9  exceeds  3  by  6. 

(-  2)2  is  4  ;       4  exceeds  -  2  by  6. 


148  ALGEBRA 

109.  An  equation  like  n^  —  n  —  6  =  0  is  called  a  Quadratic 
Equation  or  an  Equation  of  the  Second  Degree. 

Other  examples  are  :  4  x^  —  9  =  0, 

and  Sy^-^  =  -y-\-9. 

Notice  that  the  equation  has  only  one  unknown ;  that  this  unknown 
does  not  appear  in  the  denominator  of  any  fraction ;  that  it  does  appear 
with  exponent  2  ;  that  it  may  or  may  not  appear  with  exponent  1. 

Every  quadratic  equation  has  two  roots,  just  as  the  equation 
in  §  108. 

110.  Solution  of  Equations  by  Factoring  depends  upon  the 
following  numerical  fact. 

If  one  of  the  factors  of  a  product  is  zero,  the  value  of  the 
product  is  also  zero. 

Thus,  3x0  =  0;(-6)x0  =  0;2x0x(-3)=0x(-3)=0. 
Example  1.     Solve  the  equation  4  ic^  —  9  =  0. 
Solution  :  1.    Factor  :  (2  x  -  3)  (2  x  +  3)  =  0. 

2.  If  2  X  -  3  =  0,     then  (2  x  -  3)  (2  x  +  3)  =  0. 

2x-3  =  0,     if2x  =  3orxi=  +  |. 

3.  If  2  X  +  3  =  0,     then  (2  x  -  3)  (2  x  +  3)  =  0. 

2x  +  3  =  0,     if  2x=-3,  or  x=-|. 

4.  The  roots  of  the  equation  are  +  f  and  —  f . 

5.  Check  :  Does  4(f  )-2  -  9  =  0? 

^    9 
Does  ^.-- 9  =  0?  ie.  9-9  =  0?    Yes. 


Does  4(-|)'2- 9  =  0? 

1     9 
Does  <  .  ::_  9  =  0  ?     i.e.  9-9  =  0?    Yes. 

Rule.  —  To  solve  an  equation  by  factoring : 

1.  Transpose  all  terms  to  the  left  member. 

2.  Factor  the  left  member  completely. 

3.  Set  each  factor  equal  to  zero,  and  solve  the  resulting  equations. 

4.  The  roots  obtained  in  step  3,  are  the  roots  of  the  given  equation. 
Check  by  substitution  in  the  given  equation. 


SPECIAL  PRODUCTS  AND   FACTORING  149 

,     Example  2.    Solve  the  equation  -—  —  —  =  —. 

o       J       o 

Solution:  1.    Mg:*  2wi2-3m  =  36. 

2.  Sss  :  2 1»2  _  3  w  -  85  =  0. 

3.  Factor:  (2  w  +  7)(»»  -  5)  =  0. 

4.  2  m  +  7  =  0  if  w  =  —  J. 

m  —  5  =  0  if  7rt  =  +  6. 

5.  The  roots  of  the  equation  are  +  5  and  —  J. 
Check  by  substitution. 

EXERCISE  71 
Solve  by  factoring  the  following  quadratic  equations. 

1.  ar'-12.T-}-32  =  0.  11.    x'-5x  =  0. 

2.  y'-6y  =  55.  12.   Sc^-2c  =  0. 

3.  n^  =  6S-2n.  13.   8a--5a-3  =  0. 

4.  m2-18m  =  -72.  14.   4p^-^Sp  =  21. 

5.  ar'-21a;  +  110  =  0.  15.    24. i^ -\- 2 r  =  35. 

6.  c2  =  2-c.  16.   4ar^  =  8a;-3. 

7.  (p-7d-^G=0.  17.   Sw^  =  -iiv-l. 

8.  ^2-21  =  4^.  18.    3?^^2^i-§r=l. 

9.  9m2-4  =  0.  19.    2'--^  2  +  1  =  0. 
10.    36x^-25  =  0.                          20.    x^  +  jx  +  l=0. 

111.    A  Literal  Equation  is  one  in  which  some  or  all  of  the 
known  quantities  are  represented  by  letters ;  as, 

2x-j-a  =  hx'-10. 

Example  1.     Find  two  numbers  whose  difference  is  a,  and 
whose  product  is  6  times  the  square  of  a. 

Solution:  1.   Let  a;  =  the  larger  number. 

2.    Then  x  --  a  =  the  smaller  number. 

*  For  the  symbol  M^  see  ^  42. 


150  ALGEBRA 

3.  .-.  x{x-a)  =  Qa\ 

4.  .♦.  a;2-aa;-6a2=0. 
6.           .-.  (ic-3a)(ic  +  2a)=0. 

.6.  .*.  x  =  3  a  or  a;  =  —  2  a,  the  larger  number. 

Check  :  If  x  =  ^a,  then  3 a  —  a  =  2 a,      the  smaller  number  ; 

and  3  a  •  2  a  does  equal  6  a^. 

If  ic  =  —  2  a,  then  — 2a  —  a=— 3  a,   the  smaller  number  ;  and 

—  2a-  —  3  a  does  equal  +  6  a^. 

Each  solution  is  correct,  no  matter  what  a  may  be. 

Thus,  if        a  is  5,    3  a  is  15,   2  a  is  10,    and  15  x  10  is  150  ; 
also  Q.a^  =  150. 

Example  2.     Solve  the  equation  ^-^  =  5c2. 

Solution  :  1.  ?!  _  §_^  =  5  c2. 

2         2 

2.  Ma:  x'^  -Zcx-  10c2  =  0. 

3.  Factoring  :     (a;  -  5  c)  (x  +  2  c)  =  0. 

4.  If  a;  —  5  c  =  0,  then  a;  =  5  c. 

5.  If  X  +  2  c  =  0,  then  a;  =  -  2  c. 
Check  by  substitution. 


EXERCISE  72 
Solve  the  following  equations  for  x : 


1. 

Q^-\-2ax-35a'  =  0.\ 

9. 

9x^  =  4:Xv-hlSv^ 

2. 

16x2-96^  =  0. 

10. 

7x^  +  41xk  =  6k\ 

3. 

x'-\-23mx-^130m'  =  0. 

11. 

x^              3  a"     ^ 
ax — -  =  0. 

2                2 

/»»2 

4. 

25a^  =  9c2. 

5. 

2x'-^7ax-\-Sa'  =  0. 

12. 

-  =  5p'-2px. 
4 

6. 

7. 
8. 

7a^-10bx-]-Sb'  =  0. 
6x'-\-7cx  =  5c^ 

13. 
14. 

X    ,  n  9     0  ex 

SPECIAL   PRODUCTS   AND   FACTORING  151 


15.   ar  —     o     =  ^  ^''  ^^-    ~  ~ 


io 

7xt 
'  15 

^6* 

2 

5ra; 
6 

11  r* 
3 

6x2 

4xs 

3^ 

:0. 

5 

5 

2 

16.    bar =  m\  19. 


15       3  ^5 


112.  AVheii  solving  a  problem  whose  solution  leads  to  a 
quadratic  equation,  two  sets  of  results  are  obtained.  In  some 
cases,  both  sets  satisfy  the  conditions  of  the  problem;  in  other 
cases,  only  one  set  satisfies  the  given  conditions. 

Example  1.  Find  three  consecutive  odd  numbers  such  that 
when  the  product  of  the  first  and  third  is  increased  by  twice  the 
square  of  the  second,  the  sum  equals  23. 

Solution  :  1.     Let  x  =  the  smallest  number. 

2.  .*.  X  +  2  and  a;  +  4  are  the  other  two  numbers. 

(For  example,  if  3  is  the  smallest,  3  +  2  or  5  and  3  +  4  or  7  are  the 
other  two.) 

3.  .-.  a-(a;  +  4)+2(x  +  2)2  =  23. 
.-.  a;2  +  4  X  +  2(a;2  +  4  a;  +  4)  =  2.S. 

.-.  3  x2  +  12  X  +  8  =  23. 
.-.  3  x2  +  12  X  -  15  =  0,  or  x2  +  4  X  -  6  =  0. 
.-.  (x  +  6)(x-  1)=0. 
x  =  — 6orx=  +  l. 

4.  When  x=  -  6,  x  +  2  =—  3,  and  x  +  4  =  —  1. 
When  x=  +  l,  a-  +  2  =  3,      and  x  +  4  =  6. 

The  solutions  are  —  5,  —  3,  and  —  1  ;  and  1,  3,  and  6. 
Check:  (-5)  .  (-1)  + 2(-3)2  =  5  +  18=23.     1  •  6+2  •  32  =5+18=23. 
In  this  case,  both  solutions  are  satisfactory. 

Example  2.  Determine  the  base  and  altitude  of  a  triangle 
when  the  area  is  110  square  inches  and  the  base  exceeds  the 
altitude  by  9  inches. 


158  ALGEBRA 

Solution  :  1.     Let  a  =  the  number  of  inches  in  the  altitude, 

2.  .*.  a  +  9  =  the  number  of  inches  in  the  base. 

3.  ...  ei±±3  =  the  area. 

2 

.  a(a  +  9)  ^  120 
2 
.-.  a2  +  9  05  _  220  =  0.  T+g" 

.-.  (a  +  20)(a- ll)z:z0. 

.-.  a  =  11,  or  -  20. 

4.  When  a  =  11,  the  base  is  +  20  and  the  area  is  (11  x  20)  -j-  2  or  110. 
This  satisfies  the  conditions  of  the  problem. 

When  a  =  —  20,  the  result  can  have  meaning  only  if  we 
have  triangles  with  negative  altitude.  In  such  cases  we  agree 
to  take  only  the  positive  solution. 

EXERCISE  73 

1.  Twice  the  square  of  a  certain  number  equals  the  excess 
of  15  over  the  number.     Find  the  number. 

2.  Find  three  consecutive  numbers  whose  sum  is  equal  to 
the  product  of  the  first  two. 

3.  Divide  18  into  two  parts  so  that  the  sum  of  the  squares 
of  the  parts  shall  be  170. 

4.  Find  two  numbers  whose  sum  is  7  and  the  sum  of  whose 
cubes  is  133. 

5.  Determine  the  base  and  altitude  of  a  triangle  such  that 
the  area  shall  be  15  square  feet  and  the  altitude  shall  be  7  feet 
less  than  the  base. 

6.  Central  Park  in  New  York  covers  an  area  of  about  800 
acres.  Its  length  exceeds  its  width  by  2  miles.  Find  the 
dimensions  of  the  park.     (A  square  mile  contains  640  acres.) 

7.  A  merchant  sold  goods  for  $  18.75  and  lost  as  many  per 
cent  as  the  goods  cost  dollars.     What  was  the  cost  ? 


SPECIAL  PRODUCTS  AND  FACTORING  153 

8.  The  length  of  a  certain  rectangular  farm  is  three 
times  its  width.  If  its  length  should  be  increased  by  20  rods, 
and  its  width  by  8  rods,  its  area  would  be  trebled.  Of  how 
many  square  rods  does  the  farm  consist  ? 

9.  The  standard  size  city  lot  in  'parts  of  Chicago  is  five% 
times  as  long  as  it  is  wide.  The  lots  in  parts  of  Indianapolis 
are  10  feet  wider  and  5  feet  longer  than  those  in  Chicago. 
Three  times  the  area  of  a  Chicago  lot  exceeds  twice  the  area  of 
an  Indianapolis  lot  by  275  square  feet.  Find  the  dimensions 
of  the  lots  in  both  cities. 

10.  An  architect  who  has  made  plans  for  a  house  with  a 
base  30  x  42  feet  finds  that  he  must  reduce  the  size.  By  what 
equal  amount  must  he  reduce  the  two  dimensions  of  the  house 
in  order  to  make  the  area  of  the  new  base  925  square  feet  ? 


IX.   HIGHEST   COMMON   FACTOR  AND  LOWEST 
COMMON   MULTIPLE 

113.  A  monomial  is  said  to  be  Rational  and  Integral  when 
it  is  either  an  arithmetical  number,  or  a  single  literal  number 
with  unity  for  its  exponent,  or  the  product  of  two  or  more 
such  numbers. 

Thus,  3,  a,  2  a^bc^  are  rational  and  integral. 

114.  The  Degree  of  a  rational  and  integral  monomial  is  the 
sum  of  the  exponents  of  its  literal  factors. 

Thus,  a^bc^  is  of  the  eighth  degree. 

116.  A  polynomial  is  said  to  be  rational  and  integral  when 
each  term  is  rational  and  integral ;  as,  2  a^6  —  3  c  +  d^. 

The  degree  of  a  rational  and  integral  polynomial  is  the 
degree  of  its  term  of  highest  degree. 

Thus,  2a^b  —  Sc-\-  d^  is  of  the  third  degree. 

116.   Recall  the  definition  of  prime  factor  of  an  expression 
(§  87),  and  common  factor  (§  11)  of  two  or  more  expressions. 
Thus  the  prime  factors  of : 

(a)  6  m2(x2  -  1)  are  2,  3,  w,  m,  (x  -  1),  and  (x  +  1). 

(b)  9  m\x^  -  1)  are  3,  3,  m,  w,  w,  (x  -  1),  and  (x^ -\- x  +  1). 
Common  factors  of  (a)  and  (6)  are  3,  m,  m,  and  (a;  —  1). 

EXERCISE  74 
Select   the   common   factors   in   the   following   sets  of   ex- 
pressions:  ^^   5.mn'(x  +  l). 

3  •  5  •  m^n(x  —  l). 

2.    2'3'af-y'(x  +  4:)(x-S). 
2'2'3'x''y\xi-A). 
164 


HIGHEST  COMMON  FACTOR  166 

3.  7  '^•7^'S^(x-\-3)(x-2)(x-l). 
3-7  '5'7^^s{x-  3){x  -  2)(x  -  1). 

4.  6.7.  m^'n^x  -  a){^  +  "O- 
6.7.7  .mV(ar  +  a)(ic'  +  a')- 

5.  3.5.7.2.11. 
2  .  2  .  3  .  5  .  13. 

6.  3.7.5.2.2.2. 
2.2.7.11. 

117.  The  Highest  Common  Factor  (H.C.F.)  of  two  or  more 
rational  and  integral  expressions  is  the  expression  of  highest 
degree  (§  114),  with  greatest  numerical  coefficient,  which  will 
divide  each  of  them  without  a  remainder. 

Example.     Find  the  H.  C.  F.  of 

6  mH\x  -  l){x  H-  2)  and  15  m^n  (x  +  l){x  +  2). 

Solution  :  1.  The  greatest  integer  which  will  divide  both  expressions 
is  5. 

2.  The  highest  power  of  m  which  will  divide  both  is  w»*. 
The  highest  power  of  n  which  will  divide  both  is  n. 

The  highest  power  of  {x  +  2)  which  will  divide  both  is  (x  +  2). 
Neither  (jc  —  1)  nor  {x  -\- \)  will  divide  both  expressions. 

3.  .  •.  The  H.  C.  F.  is  5  m^n{x  +  2). 

Check  :  5  m^r\:\x  -  l)(a;  +  2)-t-  6  mH{x  +  2)  =  n{x  -  1). 

15m%(x  +  l)(x  +  2)  -f-5m3n(x  +  2)  =  3m(3-+  1). 

Rule.  —  To  find  the  H.  C.  F.  of  two  or  more  expressions : 

1.  Find  all  of  the  prime  factors  of  each  expression. 

2.  Select  the  factors  common  to  all  of  the  given  expressions,  and 
give  each  the  lowest  exponent  it  has  in  any  of  the  expressions. 

3.  Form  the  product  of  the  common  factors  selected  in  step  2. 

Example.  Find  the  H.C.F.  of  68 (m  +  ^0^ (w - n)*  and 
85  {m  -t-  iif  (m  —  n). 

Solution  :  1.   68(w  +  n)2(w  -  n)*  =  2  •  2  •  17  •  (m  +  ny(m  -  n)*. 

86(7n  +  w)3(m  -  n)  =  5  •  17  •  (in  +  n)8(m  -  n). 
2.    .  •.  The  H.  C.  F.  =  17(m  +  ny\m  -  n). 


166  ALGEBRA 

EXERCISE  75 
Find  the  highest  common  factor  of : 

1.  16  and  56.  6.  14  x^y,  21  xy\  and  35  a^y\ 

2.  64  a  and  96  b.  7.  15  m^<  45  mV,  and  25  miiK 

3.  72  a;2  and  27  xy.  8.  12  a^,  18  x^^/',  and  24  a^y. 

4.  5  a^62  and  2  a^ft^  ^    16mV,56mV,and88mV 

5.  20  a^^  and  15  xy':  10.  18  j-^s,  27  r^^,  and  45  rsl 
11.    (a  +  6)(a-&)  and2a(a  +  6). 

-     o  12.  3ir(a;4-2/)(aJ-2/)  and  2(x  +  y)(x-y)(x'-{-y^. 

"'■''    *  13.  (aj  +  iy(x  -  3)  and  (a?  +  l)2(a;  +  2). 

14.  (r  -\-  sf  (r  -  sf  and  (r  +  s)^  (r  -  s)l 

15.  3(a;-2  2/)(a;-h2^)  and6(a;-2  2/)'. 

16.  2  ««a;  H- 4  a^ar^^- 2  aa:^  and  3  a^a;  + 3  aa;*. 

Solution  :  1.   2  a%  +  ia'^x^  +  2ax^  =  2  ax{a^  +  2ax  +  x^) 

=  2  ax(a  +  x){a  +  x). 
3  a^a:  +  3  ax*  =  3  ax(a^  +  a;^) 

=  3  ax(a  +  a;)(a2  _  ax  +  a;2). 

2.   The  H.  C.  F.  =  a  .  X  •  (a  +  ic)  =  ax(a  +  a;). 

17.  a2-62anda2_2a6  +  62. 

18.  a;-+  2a;-24,  scr-  14a;  +  40,  and  a;2-8a;  +  16. 

19.  2r2_7r-h6and6r2_llr4-3. 

20.  a:»-27andar^-lla^  +  24. 

21.  m^  — 8  m^  and  m^  +  2m  +  4. 

22.  6  a^b'-~  15  a^b^sind  12  a'b-\- 21  a^bK 

23.  3a3  4-192and  a2-7a-44. 

24.  3  x^  -16xy-\-  5  y^  and  x"-  -{-lOxy-  75  y^ 

25.  27  a»  +  8  5^  9  a-  -  4  6^  and  9  a^  + 12  a&  4-  4  ^^l 


LOWEST   COMMON  MULTIPLE  157 

LOWEST  COMMON  MULTIPLE 

118.  A  Multiple  of  a  number  is  any  number  whieh  contaitis 
the  given  number  as  an  exact  divisor.     Thus : 

(a)  Some  multiples  of  3  are  (3,  9,  12,  and  30. 

(6)   Some  multiples  of  (x  +  y)  are  2(x  +  y)  and  (x  +  y)(x  —  y). 

119.  One  number  may  be  a  multiple  of  two  or  more  different 
numbers.     Thus : 

(a)  24  is  a  multiple  of  2,  3,  4,  6,  8,  and  of  12. 
(6)  5  a^^bc  is  a  multiple  of  5,  a=^,  6,  c,  and  of  a. 
(c)  3<j  m(x  4-  y)(a'  —  j/)  is  a  multiple  of  3,  m,  (x  -f-  y),  and  of  (x  —  y). 

A  Common  Multiple  of  two  or  more  numbers  is  a  multiple  of 
each  of  them ;  it  can  be  divided  exactly  by  each  of  them. 

120.  In  arithmetic,  it  is  necessary  at  times  to  find  the 
smallest  number  which  is  a  common  multiple  of  two  or  more 
numbers.  Thus,  30,  45,  and  00  are  all  common  multiples  of 
3  and  5  ;  but  the  lowest  common  multiple  of  3  and  5  is  15. 

Similar  necessity  arises  in  algebra. 

121.  The  Lowest  Common  Multiple  of  two  or  more  rational 
and  integral  expressions  is  the  expression  of  lowest  degree 
(§  114),  with  least  numerical  coefficient,  which  can  be  exactly 
divided  by  each  of  them. 

Example.     Find  the  L.  C.  M.  of  5  a'b^  and  7  a%\ 

Solution  :  1.   The  least  number  which  will  contain  both  5  and  7  is  36. 

2.  The  lowest  power  of  a  which  will  contain  both  o^  and  a^  is  a^ 

3.  The  lowest  power  of  b  which  will  contain  both  6^  and  6*  is  6*. 

4.  The  L.  C.  M.  is  85  a^b*. 

Check  :  Does  35  a^b*  contain  each  of  the  given  numbers  ? 
36  a^b*  -  6  a^b»  =  76;   85  a^b*  -=-  7  a^b*  =  6  a\ 

Rule.  — To  find  the  L.  C.  M.  of  two  or  more  expressions  : 
1.   Find  the  prime  factors  of  each  of  the  expressions. 


158  ALGEBRA 

2.   Select  all  of  the  different  prime  factors  and  give  to  each  the 
highest  exponent  with  which  it  occurs  in  any  of  the  expressions. 
•  3.  Form  the  product  of  all  of  the  factors  selected  in  step  2. 

Example.     Find  the  L.  C.  M.  of   25  a^h  (a  -\-  b)\a  —  h)  and 

Solution  :  1.    25  a^h{a  +  6)3(a  -  ft)  =  52  a'^h{a  +  hy{a  -  b). 

15  a%\a  +  b)\a  -  6)3  =  3  •  5  a^b\a  +  b^^a  -  by. 

2.  3  occurs  with  1  as  its  highest  exponent. 

5  occurs  with  2  as  its  highest  exponent. 
a  occurs  with  3  as  its  highest  exponent. 

6  occurs  with  4  as  its  highest  exponent, 
(a  +  b)  occurs  with  3  as  its  highest  exponent, 
(a  —  b)  occurs  with  3  as  its  highest  exponent. 

3.  The  L.  C.  M.  =  3  .  52  a^b\a  +  by(a  -  by. 

Check  :  Does  the  L.  C.  M.  contain  each  of  the  expressions  ? 

75  a^b\a  +  by  (a  -  by---  25  a26(a  +  by  (a  -  6)  =  3  ab^(a  -  by. 
75  a8M(a  +  by(a  -  by-^  15  a^b\a  +  by^a  -  by=  5(a  +  b). 

EXERCISE  76 

Find  the  L.  C.  M.  of  the  following  and  obtain  the  quotient 
when  the  L.  C.  M.  is  divided  by  each  of  the  numbers : 

1.  5  and  7.  3.   24  and  30.  5.   15,  21,  and  33. 

2.  12  and  20.  4.   12  and  54.  6.   20,  27,  and  90. 

7.  3  ab  and  7  a^b.  12.    15  a^y,  30  xif,  and  60  xY. 

8.  12  oc^if  and  48  xy*.  13.    5  ?-m^,  15  rm^,  and  21  r^m. 

9.  12  m»  and  15  mn\  14.   24 p^  32 p%  and  12^. 

10.  24  a%*  and  16  a^fe*.  15.    32  wv,  16  id^v^,  and  64  w\ 

11.  14  rV  and  35  r^.  16.   44  xy,  33  yz,  and  12  xz. 

17.  (a  +  6)(a-6)  and  (a4-?>)^' 

18.  2  a(m  4-  a)  and  6  a\m  +  a)(m  —  a). 

19.  (a;  +  3)(a;-2)  and  (a;-2)(a;-3). 


LOWEST   COMMON  MULTIPLE  159 

20.  3(r  -f  s)(r  -  s)  and  2(r  -f  s)(r  -  t). 

21.  (a-4)(a-3)  and  (a-4)((X-5). 

22.  (1  -  xy,  (1  -  x)(l  4-  x),  and  (1  -  xf. 

23.  (2-3  a;)2,  3(2  -  3  x)(2  -f-  3  x),  and  (2  -f-  3  xf. 

24.  (2  -  a;)(3  -  x),  (3  -  a;)(4  -  x),  and  (4  -  x)(2  -  x). 

25.  4  ic^  —  4  wi^,  6  a;  +  6  m,  and  3  ar*  —  3  7/i\ 

Solution  :  1.  4x^  —  im^  =  i(x^  —  m^)  =  4(x  -  m)  (x-\-m). 

6x  +  6m  =  6(a:  +  m). 
3  x8  -  3  w8  =  3(a^-»/i»)  =3(r- w)(3:2  +  mx  +  m^. 

2.  The  L.  C.  M.  =  12(a;  -  m)  (x  +  w)  (a:2  +  mx  +  m^). 

3.  L.  C.  M.  -?-  4(x  -  m)  (X  +  m)  =  3(x2  -f  wix  +  m2) . 

L.  C.  M.  ^  6(x  +  m)  =  2(x  -  rn)  (x2  +  »nx  +  m*). 
L.  C.  M.  -^  3(x  -  w)(x2  +  mx  +  m^)  =  4(x  +  m). 

26.  r2-16and7-2  4-llr-}-28. 

27.  a^  +  2  ttic  4-  a^  and  a^  +  xl 

28.  Hx^y  —  y  £^nd  22  ar^z  —  9  a;2  —  z. 

29.  62_i264.35,  62-}- 2  6  -  63,  and  6^-3  6  - 108. 

30.  4a:2_25and4a:2_20a;-|-25. 

31.  3  m^-  6  m  -  72,  4  m^  +  8  m  -  192,  and  2  m'-  24  m  +72. 

32.  9n2-27n  +  8and3n2-2n-16. 

33.  l-|-27a:»andl-5a;-24ar^. 

34.  ar^  +  a;  -  42  and  x2  _  a;  _  ^q 

35.  a*'  —  a  and  a^  —  9  a*  —  10  a. 


X.   FRACTIONS 

122.  The   quotient  of   a  divided  by  b  is  written   -•     The 

b 

expression  -  is  called  a  Fraction ;  the  dividend,  a,  is  called  the 

Numerator,  and  the  divisor,  b,  is  called  the  Denominator.  The 
numerator  and  denominator  are  called  the  Terms  of  the  frac- 
tion. -  is  read  "  a  divided  by  bJ'  The  denominator,  b,  must 
never  be  zero  (§  64). 

REDUCTION  OF  FRACTIONS  TO   LOWEST  TERMS 

123.  A  fraction  is  said  to  be  in  its  Lowest  Terms  when  its 
numerator   and    denominator  have  no  common  factor   except 

.  unity.     Thus : 

(a)   -  ,   -,   ^-i-^  are  in  their  lowest  terms. 
3     b     X  —  y 

(.b)   i~ ,  — T" )    ;  ,     ^1    are  not  in  their  lowest  terms. 

124.  To  reduce  a  fraction  to  lowest  terms,  a  principle,  easily 
illustrated  by  arithmetical  fractions,  is  used. 

If,  in  the  fraction  y ,  both  terms  be  divided  by  4,  a  new  fractionf  is 
obtained.  But,  since  -2/  =  3,  and  f  =  3,  therefore  ^^^  =  |.  The  value  of 
the  fraction  is  not  changed  ;  its  form  is  changed. 

Rule.  —  If  the  numerator  and  denominator  of  a  fraction  are  both 
divided  by  the  same  number,  the  value  of  the  fraction  is  not  changed. 

Example  1.     Reduce  to  lowest  terms 

40  a^b'c 

Solution:  1.  gjoB^  ^2^  ■  3  .  gB .  52  .  c 

40  a^b-^c     23  .  5  .  a^  .  62  .  c 
2.   Divide  both  numerator  and  denominator  successively  by  their  com- 
mon factors,  23,  a^,  b^,  and  c. 

160 


FRACTIONS 

24  a^b^c 

I 

a 

H^ 

1 

3a 

AOa^b-'c 

1 

1 

1 

1 

5 

161 


Then, 


Note  1.  Dividing  by  all  of  the  common  factors  is  equivalent  to  dividing 
both  numerator  and  denominator  by  their  highest  common  factor. 

Note  2.  Tlie  process  of  removing  a  common  factor  from  numerator  and 
denominator  of  a  fraction  is  called  cancellation.    It  depends  upon  the  rule  in 

§m. 

Note  3.    It  is  wise  to  write  the  quotients  1,  a,  etc.,  as  they  are  obtained. 

Example  2.     Reduce  to  lowest  terms  -I-  o4  6 

9a2-f  24a6-}-166« 
1 
Solution-         27 a^  +  64 6«        _  QloM-^) (0 a'^ -  12 q5  -f  16 ft^) 
■  9a2  +  24a6  + 16  6-^  C2^z-h<r5)(3a  +  4  Z>) 

1 
^9a2-12a&  +  16  62 
(3  a  +  4  &) 
Here,  the  numerator  and  denominator  are  both  divided  by  (3  a  +  4  6). 
Check  :  These  examples  may  be  checked  by  substitution.     It  is  im- 
portant to  remember  that  the  original  fraction  and  the  simplified  result 
are  equal  for  all  values  of  the  literal  numbers  except  such  as  make  the 
denominator  zero.     (§  122.) 

Rule.  —  To  reduce  a  fraction  to  lowest  terms  : 

1.  Express  numerator  and  denominator  in  terms  of  their  prime 
factors. 

2.  Divide  both  numerator  and  denominator  by  all  of  their  common 
factors,  i.e.  by  their  H.  C.  F. 

125.  Errors  in  Reducing  Fractions.  One  common  error  occurs 
in  reducing  fractions  such  as 

3win2(x +  2/)      ^^ifitC0  {^[.^Y^' 
Pupils  sometimes  think  that  the  result  is   0,  because  all 
factors  have  been  cancelled.     If,  as  suggested,  the  quotients 
are  indicated,  this  danger  will  be  avoided ;  thus, 

1  11       1 

3mn2(x  +  j/)      '^jntn^  {xA^)      1 
1111 


162  ALGEBRA 

Another,  and  more  common,  error  is  illustrated  in  the  fol- 
lowing faulty  solution : 

1      1 

2a  +  6     aiJk-r5     2  +  1 


ah  ;ib  \ 

11 


=  3. 


Testitfora  =  2,  6  =  2;  ?A±^  =  idL2  =  6  ^  1 
a6  4         4     2 

The  error  is  in  dividing  one  part  of  the  numerator  by  a  and 
another  part  of  it  by  h.  Neither  a  nor  6  is  a  factor  of  the  nu- 
merator. They  cannot  be  canceled  in  this  problem.  The  frac- 
tion is  already  in  its  lowest  terms. 

EXERCISE  77 

Reduce  the  following  fractions  to  their  lowest  terms :    . 

^     12  5^.  ^^r'sf 

15'  *    Sxf  '    eSr^s^^^- 

36*  '    42mV'  *    40xy*z' 

^     3ab'c(x-^y)  '  ^^     2af-2y\ 

2abc{x-\-y)  '    5a^  —  5y^ 

8     12(c^-3)(a  +  3)  ^^         Sam' -3  an'' 

4(a-f3)         *  *    3m2  +  6mnH-3w2* 

.     18(2r-gy(r-f^)  ^g  «'  +  ^'^ 

*    9(2r-s)(r  +  s)2  '    a^c - 2 a6c - 3 fe^c 

^^^    a^  +  7a-flO  17,         8a^-125 

(x2-^4a  — 5  '    2  ax' -\- ax  — 15  a 

m^  +  m  —  56  -g     ma;^  —  ma;  —  12  m 

■    m^  -  m  -  42  *  '     3  ic'^  +  13  a;  + 12  ' 

a^_9a;4-18  ^^     169-^  +  47-^  +  1. 

3a^^-3a;-36*  '         64r«-l 

,,     a2-lla6  +  2862  ^^       a3-8a2  +  12a 


^_14a6  +  4962  Sa^-- 60a +  180 


FRACTIONS 


168 


21. 


22. 


23. 


^-f 


aa^  —  2  aacy  -\-  ay^ 

a^-64 
a%  — 16  m 

(«^-4)(a-l) 


24. 


25. 


4  7n^  -f  16  m??  4- 15  n^ 
6  m^  —  mn  —  15  n^ 

15  a;^y  +  10  x^y^ 


6a:«2/'  +  4ar^/ 
26.        ^^-^y^      . 


126.  Signs  in  Fractions.  A  fraction  is  an  indicated  quotient. 
Its  sign  is  governed  by  the  law  of  signs  for  division  (§  67). 
Thus: 


From  1  and  2,  it  is  clear  that 

•4-4 
From  1  and  3,  it  is  clear  that 
4-12^ 
+  4 


12 


+  4 
-12 


=  -3. 

=  -h3. 


+  12 

-4' 
-12 

+  4'J 

since  +3  =  — (-  3).. 


Rule.  —  If  the  sign  of  one  term  of  a  fraction  is  changed,  the  sign 
of  the  whole  fraction  must  be  changed. 


From  1  and  4,  it  is  clear  that 

+  12^-12 
+  4        -4 


,  since  both  equal  +  3. 


Rule.  —  If  the  signs  of  both  terms  of  a  fraction  are  changed,  the 
sign  of  the  fraction  must  not  be  changed. 

Changing  the  signs  of  an  expression  is  accomplished  by 
multiplying  it  by  —  1. 

Example  1.     Reduce  to  lowest  terms  : 
12  +  2x-2ar»* 


164  ALGEBRA 

Solution  :  Multiply  the   denominator  of   the  given  fraction  by  —  1, 
thus  changing  its  signs ;  also,  change  the  sign  of  the  fraction. 

1 
x2-9         ^  x2-9         ^       (aL^^3)(x  +  3)  ^        a; -f  3 

12  +  2  X  -  2  x-2  2  x2  -  2  a:  -  12  20-^')  (x  +  2)  2{x  +  2)  ' 

1 
a:2-9  1-9  -8         2 


Check.     Let  x  —  \,     Then 
also 


12  +  2x  -  2x2       12  +  2-2       12  3' 

X  +  3  1+3  4  2 


2(x  +  2)  2(1  +  2)  6  3 

The  value  of  x  selected  must  not  make  any  denominator  zero  (§  64). 

Example  2.     Reduce  to  lowest  terms: 

(9  -  m^) 


Solution  :  Multiply  the  numerator  by  —  1,  thus  changing  its  signs  ; 
also  change  the  sign  of  the  fraction. 
^     _        9-^2         ^  w2  -  9 


w2  -  7  m  +  12         m2  —  7  m  +  12 
1 
2     ^  (m^-^)(m  +  3)  ^  (m  +  3) 
(m  -  4)  (W2^-^)      (m  -  4)  ' 
1 


EXERCISE 

78 

Reduce  to  lowest  terms  : 

42/2-0^ 

6. 

27-^ 

'   2x'-7xy-]-6y^ 

e-9 

Sx-371 

7. 

a?-l^ 

"'  f^xy-^x" 

64 -ar^ 

3          «'  -  ^^ 

8. 

at^  -  as" 

24  -  2  a  -  tt^ 

2  as^  +  4  ars  —  6  ar^ 

^          3  7?r  -  3  71^ 

9. 

3c«_  6c2rZ  +  3cc?2 

n^  +  4  mri  —  5  wi^ 

bd^-bc" 

a^2  ^  a;  -  6 

10. 

18  ma?2  _  8  ^^2 
24  W2/3  -  81  nx^ 

FRACTIONS  165 


TO  REDUCE  A  FRACTION  TO  AN  INTEGRAL  OR  MIXED 
EXPRESSION 

127.  An  Integral  Expression  is  an  expression  which  has  no 
fractional  literal  part ;  as,  a^  —  2  ah  +  fr,  or  |  ah-. 

An  integral  expression  may  be  considered  a  fraction  whose 

denominator  is  1 ;  thus,  a  +  6  is  the  same  as  —^ — 

128.  A  Mixed  Expression  is  an  expression  which  has  both 
integral  and  fractional  literal  parts  ;  as,  a  +  -  or  x  +  ^         • 

C  7J-Z 

Rule.  —  A  fraction  may  be  changed  to  an  integral  or  mixed  expres- 
sion by  performing  the  indicated  division. 

Example  1.    Reduce — ~  "■  to  a  mixed  expression. 

3  X 

Solution  :  Using  the  method  of  short  division  (§  70), 

6^+16x-2^6^      150.      -j^,  _2_.    Ans. 

3x  3x       3x       8«  3x 

Example  2.   Reduce ~ — -— ^  ^  ~  -  to  a  mixed   ex- 

4  af'  +  3 
pression. 

Solution  :  Using  the  method  of  long  division  (§71), 

3  a: -2 


4t«a  +  3|l2a:*t- 

-8x--J-h4x- 

-5 

12a:8 

-l-9a; 

.8a:2-5a-- 

-6 

■8aa 

-6 

-5ar-f  1 

12x8 

_8x2  +  4a;-5 

4a:'-i  +  3 

=  3a;-2- 

=  3  a;  -  2  + 

4a;2-f  3 

-6r 

r  +  1 

+  3 

The  first  term  of  the  numerator  of  the  fraction  in  the  resalt  is  negative. 
Change  the  signs  in  the  numerator  by  multiplying  the  numei-ator  by  —  1, 
and  also  change  the  sign  of  the  fraction.     See  §  126. 


166  ALGEBRA 

EXERCISE  79 

Reduce  the  following  to  mixed  expressions : 

1.    W-  2.    «.  3.    ^K  4.    ^\K 

5    12a^-16ft  +  7  ^     15j9^H-12iJ-4 

4a  '  *  3i)  * 

15m3-6m2  +  3m-8  „    30a;«  + Sa^- 15a^- 7 

o.     ■ 


a?  +  2f 

x-y 

a«  +  8  6« 

13. 

x-y 

14    a^4-^-^y+6y^ 
a-2h  '        x-2y 

16    3a«  +  8a^-7 
4a-l  *     a2-2a-3 

TO  REDUCE  FRACTIONS  TO  THEIR  LOWEST  COMMON 
DENOMINATOR 

129.  In  arithmetic,  fractions  having  a  common  denominator 
may  be  added  or  subtracted  without  difficulty.     Thus : 

Fractions  which  do  not  have  the  same  denominator  must  be 
changed  to  equal  fractions  having  a  common  denominator. 

(a)  -^  +  I  =  A  +  T%  -  H,  for  f  =  j%. 
In  algebra,  also,  fractions  which  do  not  have  a  common  de- 
nominator must  be  changed  to  a  common  denominator  before 
they  can  be  added  or  subtracted. 

130.  To  change  fractions  to  a  common  denominator,  a  prin- 
ciple, easily  illustrated  arithmetically,  is  used. 

If  both  terms  of  the  fraction  |  are  multiplied  by  4,  the  result  is  ^. 
I  =  ^  since  both  equal  3. 

The  value  of  the  fraction  is  not  changed  ;  its  form  only  is  changed. 


FRACTIONS  167 

Rule.  —  If  the  numerator  and  denominator  of  a  fraction  are  both 
multiplied  by  the  same  number,  the  value  of  the  fraction  is  not 
changed. 

131.  Two  fractions  which  have  the  same  value  but  different 
form  are  called  Equivalent  Fractions. 

Thus,  -  and  —  are  equivalent  fractions ;   they  differ  in  form,  but 
b  mb 

have  the  same  value,  since  the  second  is  obtained  by  multiplying  both 

terms  of  the  first  by  m. 

132.  The  Lowest  Common  Denominator  (L.  C.  D.)  of  two  or 
more  fractions  is  the  lowest  common  multiple  of  their  denom- 
inators. 

Example  1.     Reduce  to  their  lowest  common  denominator 

i!?^  and  ^^. 
2  ab'         3  a'b 

Solution  :  1.    The  L.  C.  M.  of  2  ah^  and  3  a'6  is  6  aW.  (§  121) 

2.  To  change  the  denominator  2  ab^  into  0  a^ft^,  we  must  multiply  2  ab^ 
by  3  a^,  that  is  by  6  a^b'^  -r-  2  ab'^.  In  order  not  to  change  the  value  of  the 
fraction,  the  numerator  3  mx  must  also  be  multiplied  by  3  a^. 

Then,  Sn^^Sa^.Zmx^^Qa^mx^  . 

3.  For  the  second  fraction,  6  a^b^  -4-  3  a^ft  =  2  6.  Multiply  both  nu- 
merator and  denominator  by  2  b. 

Then,  ^^  =  UilAm.  =  lOfcnj^ .  (§  130) 

Rule.  —  To  reduce  fractions  to  their  lowest  common  denominator: 

1.  Find  the  prime  factors  of  the  denominators. 

2.  Find  the  L.  C.  M.  of  the  given  denominators ;  this  is  the  L.  C  D. 

3.  For  each  fraction,  divide  the  L.  C.  D.  by  the  given  denominator ; 
multiply  both  numerator  and  denominator  by  the  quotient. 

Example  2.     Reduce  to  their  lowest  common  denominator : 

4«     and         3a 


a'-4         a'-Ba  +  6 


168  ALGEBRA 


Solution  :  1. 


2. 


4a  4a 


a2-.4      (a-2)(a  +  2) 
3a  3a 


a^  -  5  a  +  6      (a  -  2)  (a  -  3) 

3.   TheL.  CD.  is(a+2)(a-2)(a-3). 

•  4.   L.  C.  D.  -^  (a  -  2)(a  +  2)  =  a  -  3. 

4a 4  a(a  —  3) 

"  (a-2)(a  +  2)  ~  (rt-2)(a  +  2)(a-3)' 

5.  L.  C.  D.  -  (a  -  2)  (a  -  3)  =  (a  -f  2). 

.  3  a  ^  3aCa-t-2) 

"  (a-2)(a-3)      («  -  2)(a  +  2)(a  -  3)' 

Check  :  The  final  fractions  in  steps  4  and  5  may  be  changed  into  the 
original  fractions  by  cancellation. 

EXERCISE  80 

Reduce  the  following  to  equivalent  fractions  having  their 
lowest  common  denominator : 


1-  i;  I;  |.  9- 

2-  f;  tVj  tu- 


12  6 


3. 


4. 


3a     55 
T'     6 


3  '     5  '     2  *  12. 

5_mw,  7  mp^  3_np 

5    '  13. 

14. 


6 

> 

4    ' 

2. 
a 

3    5 

• 

2x 
a 

.  3.V 

bz 

ah 

c  , 

A. 

xi 

xz 

2/2 

15. 

8.   — ;   -;   — .  16. 


2  mhi 

3mn^^  5  7nhi^ 

i^s 

^           ^ 

2m^n 

3mn''   5  77iV 

2a-5 

c.    4a-f36 

^ab 

'       12  ac 

ZX-A: 

z    6x  —5z 

4.x^ 

'       3  x'z 

2ab 

b 

a'-b'' 

a-{-b 

A  a' 

2 

4a2_s 

1'   6a'-9a 

2x 

.      3?/ 

x^2y' 

'   ^'-2?/ 

^x 

5 

5  a; -10'   2  a; -4 


FRACTIONS  169 


17.    ^ ;        ^      .  19.       «+^     -     ^-^ 


af-Qx-i-S'  a^-16  a«-2a6'  aft-f-fc^ 

18.   -AZL;   _-^ 20       ^  ^  ^ 


a  H-  5  a  +  3 


21. 
22. 
23. 
24. 
25. 


a2_a-6'   a^  +  Ta-l-lO 

a  +  36  a  —  36 


a^  _  7  a/^  _^  12  6^'   a-  -  a6  -  12  6^ 

x^—2xy-\-if^   {a-{-b)(x  —  y) 

x-\-y     ^        xy 

a-j-2  a— 3  a  4- 1 


a2-|_2a-3'  a2_3^^2'  a^~^a-6 


ADDITION  AND  SUBTRACTION  OF  FRACTIONS 

133.    Example  1.     Perform  the  indicated  addition  : 
4aH-3     l-6b' 
4  a^b  6  ab^ 

Solution  :  1.  The  fractions  cannot  be  added  because  they  do  not 
have  a  common  denominator.  By  the  methods  of  §  132,  change  the  two 
fractions  to  equivalent  fractions  having  their  lowest  common  denominator. 
2.  The  L.  C.  D.  =  12  a^b^.  Multiply  the  terms  of  the  first  fraction  by 
12  a^b^  H-  4  a^b,  or  3  b'^ ;  and  the  terms  of  the  second  fraction  by  12  a^b^ 
^  6  ab^,  or  2  a.    Then, 

3    4  g  -t-  3      1  -  6  ?)2  ^  3  feg  X  (4  g  +  3)      2  a  x  (1  -  6  S^) 
4  a-^b  6  g68  3  6^  x  4  g^6  2  a  x  (6  g&8) 

^  (12  afo2  4.9^2)      (2a-12a62) 

12  g268        "^        12  g268 
_  (12  ab^  +  9  &2)  +  (2  g  -  12  ab^) 

12  g263 
_  12  aft^  +  9  62  +  2  g  -  12  ab^ 

12  0*63 
_  9  62  +  2  g 
12aa68    * 


170  ALGEBRA 


Check  :  Let  a  =  I ;  h  =  1. 

4^4-3^7     l-6fe2^-5    ^^^7      (-5)^11. 
4a-^6        4'     6a&3  6    '  4  6  12* 

also,  9b^±2a^9_±2^n^ 

12rt253  12         12 


Rule.  —  To  add  or  subtract  fractions  : 

1.  Reduce  them,  if  necessary,  to  equivalent  fractions  having  their 
lowest  common  denominator. 

2.  For  the  numerator  of  the  result,  combine  the  numerators  of  the 
resulting  fractions,  in  parentheses,  preceding  each  by  the  sign  of  its 
fraction. 

3.  For  the  denominator  of  the  result,  write  the  L.  C.  D. 

4.  Simplify  the  numerator  by  removing  parentheses  and  combin- 
ing like  terms. 

5.  Reduce  the  result  to  lowest  terms. 

Example  2.     Simplify  ^^^^ii' -  I^^^^j . 

6  14 

Solution:    L    5a^  -  4  y  _  7  x  -  2  y  ^  7(5x- 4  y)  _  3(7  a:  -  2  y) 
6  14  42  42 

2  ^7(5a;-4y)  -Snx-2y) 

42 

^  42 

4  .  ^  14  a;  -  22  y 

42 


5. 


Check  ;  Let  x  =  1 ;  y  =  1. 


20x-ny)  ^7x-n 
42  21 


5x-4y  ^  5-4  ^  1     lx-2y  ^b^    ^^^1       5^-8^-4. 
6  6  6'         14  14'  6      14       42        21  ' 

also,  1  x-\\y  _  7-11  ^  ^  ^     r^^^  solution  is  correct. 

'  21  21  21 

Note.    In  the  first  step  of  the  solution,  the  numerator  and  denominator 
of  the  first  fraction  are  multiplied  by  7,  and  of  the  second  fraction,  by  3. 


FRACTIONS  171 

EXERCISE  81 
Perform  the  indicated  additions  and  subtractions : 

9     27     3  3     6^15 

2     ±4-?_l  9     ^-l£^lA 

'    15      5     20  '9         3  "^27 

3.   A  +  ?_i.  •  10.    ^+^_^. 

16     8      4  6      10     15 

I4.I-A.  11     6(1-5      3a  +  7 

•    14  "^21      84*  *         8      "^      12     ' 

^     5a  ,  3a  -„     2a-8^3a  +  5 

6.    ^^   -f^.  12.    -^_+--^^. 

-     6m     3  m  -„     3  7^1  + 4     2m-|-5 

*•  T'u:'  ^^'  ~i2       ^9 

„     15a;     7a;,  3a;                      ..     4a-9     3a-8 
7       u  - — .  14. 

8         4        2  9  12 

,,     5a-f  1  ,  26  +  3      7c-4 
''•     "6~  +  -8 12-' 

3a  +  4      4a-3  .  5a  +  2 
16.    _ __4.___. 

2a-36     3a  +  6     4a-56 
9  18  27       * 

6m  +  l      5m  — 2     8m  — 3     7m  +  4 
^®'    "~3  6~~"^~^  i2~* 

,-     4r-3s.6r  +  5s     5r+2s     3r-10s 


10  15  20 

«^     3t-2x     7t-Sx  ^  9t-{-4x     lOt-^lx 

iiU.       ^ 1 — — ' 


172  ALGEBRA 

21.    ?-A.  26. 


22.     — -  +  -^.  27. 


3_ 

5 

a 

2a 

2 
Sx 

42/ 

2 

7'        S         t 

.7 

X      z 

a b c_ 

2  m^     mw     5  n^ 


5a4-l  .  3a 


6a  2a 

23.  i_l+^.  28.    lQ^  +  3y     3a;  +  5y 

2  ic^i/  iC2/^ 

24.  -  +  ---.  29.    ^:zl  +  fc2_?_j_5--3^ 

a»/  2yz         3zx 

25.  -  +  -_-.  30.    2a-6^26-c_^2c- 


31. 


a6  6c  ca 

1 


Solution:  1 
2. 


cc^  -f-  a?     x^  —  X 
1  1  1 


a:2  +  a;     x^-x     x(x  +  1)      a;(x  -  1) 

(0^-1) (a^  +  1) 

ic(a;+ l)(x- 1)      a:(x  +  l)(a;  -  1) 

3  ^(a;-l)-(x  +  l) 

a;(x+l)(x-l) 

^  x—\—x—\ 


x{x-^  l){x—  1) 
-2       ^       +2 


Notice  that  in  line  2,  the  L. CD.  is  a;(x  +  l)(x—  1) ;  that  the  numerator 
and  denominator  of  the  first  fraction  are  multiplied  hy  (x  —  1) ,  and  of  the 
second  fraction  by  (a:  +  1)  ;  that  parentheses  are  used  in  lines  2  and  3  ;  that 
in  line  5,  the  signs  of  both  numerator  and  denominator  are  changed.  Check 
by  substitution. 

32.    Simplify       «  +  ^  ^"^ 


4  a^  -  9  6=^      (2  a  +  3  b)' 


Solution:  1.       «  +  ^  ^"^ 


2.    = 


a  4-h  a 


(2a-3&)(2a  +  36)       (2  a  +  36)(2  a  +  3  6) 

3     ^  (q  +  6)C2a  +  3&) C«-6)(2a-3  6) 

(2a-36)(2a  +  36)(2a  +  3&)      (2a-3  6)(2a+36)(2a+3  6) 


FRACTIONS  178 


.     ^  r2  gg  4-  6  orft  -f  3  ^2)-(2  a^-^ab  +  S  &«) 
(2  a  -  3  6)  (2  a  +  3  6)  (2  a  +  3  6) 

2  a2  +  5  a6  +  3  62  _  2  a2  +  5  aft  -3  62 
o. 


6. 


(2  a  -  3  6)  (2  o  +  3  6)(2  a  +  3  6) 
10rt6 


(2a-3  6)(2a  +3  6)(2a  +  3  6) 
Check  :  Let  a  =  1,  6  =  1. 

«  +  ^      =^  =  -^-,  '-'       ^-0^0,and-^-0^-g; 

4a2-962      -5         6'     (2 «  +  3  6)^      25        '  5  6 

also  10  «&  10         ^    10    ^      10^      2 

(2a-3  6)(2a  +  36)2     (2-3)(5)2     -25         26         5 

Notice  that  the  indicated  products  in  step  3  in  tlie  numerator  are  found 
and  inclosed  in  parentheses  in  step  4.  All  of  this  solution  should  be  done 
mentally. 

33.       ^  1      ■  41.    2-0?      2-\-x^ 

'   rn  —  l     m-\-l  2-\-x     2  —  X 

34    -^^4--^.  42    4p^  +  l      2;)-! 

•   r+a     r-3  ■    4:^^-1     2p  +  l 

35.    -i ?-.  43.    -1 i2«-6)_^ 

5m-2     2m-h3  2a  +  6      Sa^  +  ft^ 

3  6  5  6 

36. 


3a-4      5a+6 

37.     ^y         ^^ 

•    *   3y  —  X     2a;— 3y 

38       ^      I      y     - 

•   2x-\-2y     3x-3y 
1  2 


39. 


40. 


3a-.7      6a4-15 

__3 4 

4i)-6      15;)- 12* 

2 


49. 


44. 

x-^S     ar'^-27 
x-S     x'~21 

45. 

x-\.2     x-2        16 
x-2     x+2     x'-4. 

46. 

11            2  a 
a^h     a-h     a^-h^ 

47. 

1            Sx           ax 

a  —  x     a^  —  x'     a*—  x^ 

48. 

m-i-n      m  —  n        4  mn 

m  —  u      m  +71      m^  —  n^ 

5 

ic2_5a;+6     x2_,.2a._i5 


174  ALGEBRA 


KA        q+l  (^  — 4         ,       a  4- 3 

oO-   -s t: o \ —7:  + 


a2_a-6      a2-4a-f-3      a2H-a-2 
3a-|-2       ,  a  +  3  a-2 


62. 


Ba^-a-l      3a2+7a  +  2      2a2  +  3a-2 

r a 

r^  —  6  ar  +  9  a^     r^  +  4  ar  —  21  a^ 


g3  a;  +  2  2(a^-l)  a;-3 

a;24-4a;  +  3     ar^  +  a;-6     a;2-a;-2* 

-,         1  a  ,  a^  — 4 

a  +  l      a^-a  +  l      a^  +  l 

55.    Simplify  -^  +  ^^-±4- 
^     ^  a-b      b'-a' 

Solution  :  1.  Notice  that  b^  —  a^  is  not  in  the  same  order  as  a  ~h. 
Change  the  signs  of  the  denominator  and  also  the  sign  of  the  fraction 
(§  126). 


2 

3         2h+a_     3         a4-26 

a-h      b'^~  aP-     a-b      a^  -  b^ 

9 

3                a  +  2b 

a-b      (a-6)(a  +  6) 

4. 

_3(a  +  6)-  (a +  2  6) 

5. 

_3a  +  3?)-a  -26 

6. 


{a  —  b)ia  +  b) 
2rt  +  & 


a2  -  62 
Check  by  substitution. 

56.  -1^  +  -^.  59. 

57.  — ^t 1 60.   -^+    ^ 


y^  —  xy     x^  —  xy  a  + 1     1  —  a     tt^  —  1 

x-\-4:       2  X  —  5  r  r     _     r^ 

3a;-6~8-4a;*  '    r  +  2"~2-r     7^-4' 


FRACTIONS 

62. 

a            b           2 

b' 

-a' 

67. 

5a-b 

rt  -f  6     a-b     b'^  ' 

1      5a+b 

63. 

2x                  ?^x 

68. 

3a  +  l      ^-".?- 
2a  — 3 

Q-x-x'     x'-^^x 

-10 

64 

^'-i-y 

69. 

.           «'  -  b' 

a'  +  ab  +  b^ 

65. 

'-'*'-^- 

70. 

3^  +  4      «™'  +  f 
3  wi  —  4 

66. 

X+7J 

71. 

r'      rs  +  s'       ^^. 

176 


MULTIPLICATION  OF  FRACTIONS 

134.  In  arithmetic,  the  product  of  two  fractions  is  the 
product  of  their  numerators  divided  by  the  product  of  their 
denominators.     Thus,  f  x  ^^  =  ^^. 

In  algebra,  the  same  rule  is  followed. 

Example  1.     Multiply  —  by  —  • 
2  X       15  a^ 

'     Solution:  1.   ^-«  .  i^  =  ?Oax^ 
2x15  a2      30  a'^x 
2.   Reduce  to  lowest  terms : 

2  X 
20aa^_^a^_2x 
30a^x~mrth7~  Sa' 

3  a 

It  is  customary  to  cancel  the  common  factors  in  step  1  as  in 
the  following  example. 

Example  2.     Sin.plity  ?^±2a^  .  a^. 

a^  — 16  a^  —  1 


Solution  :  1. 


q-^  +  2  g  -  3     a^-4a 
a'i  -  16       '    a2  -  1 

1  1 

1  1 

_       a(a  +  3)       ^      gg  +  3a 
(a  +  4)(a  +  l)      a2  +  5a  +  4' 


176  ALGEBRA 


Check  :  Let  a  =  2. 
a2  +  2a-3_4-f4-3 

a^-4a 


also, 


4-16 

_4 
4 

-8_- 
-  1 

a2 

4-3a 

5 

-12 

-5 
12 

-4 

3    ' 

12        3  36 

5 


a2_  1 

4  +  6  10      5' 


a2  +  5  a  +  4     4+10  +  4      18      9 

Notice  that  the  factors  (a  —  1)  of  the  first  numerator  and  of  the  second 
denominator  are  each  divided  by  (a—  1),  or,  are  cancelled ;  similarly  the 
factors  (a  —  4)  of  the  first  denominator  and  the  second  numerator. 

Rule.  —  To  find  the  product  of  two  or  more  fractions : 

1.  Find  all  of  the  prime  factors  of  the  numerators  and  denomina- 
tors. 

2.  Divide  out  (cancel)  factors  common  to  a  numerator  and  a  denom- 
inator. 

3.  Multiply  the  remaining  factors  of  the  numerators  for  the  nu- 
merator of  the  product,  and  of  the  denominators  for  the  denominator 
of  the  product. 

EXERCISE  82 

Find  the  following  indicated  products  : 

,      3       21      10  .    5a'b       Jy'G       ^c'a 

A.     T~r    •    777:    *    ^:r '  o. 


3 

21 

10 

14 

20 

9 

5 

6 

12 

18 

10 

7 

^'   ^  '  ~  '  ^'  7. 


6  a^h  .   ^xy^ 
15  m^y      2  ah 

3  am"       25  W 


20  U'      3  aW 

_    B  a 

5.    — -    • 


9. 


^a      66        8c 
4  6      10  c      12  a 

10.    ^^'-^ 
m^  —  16  m 

^^     a2  +  2a-35 
6a3 

'  {a 

2a(a-3) 
+  7)(a-3) 

2oc2 

3  6a^ 

5 

b'c 

4a« 

9  6^ 

15  6^ 

21 
10 

a'' 

27  7ri'y 

'       2,0  n 

T.xy 

20  m^x 

14 

t-y 

ISn' 

x' 

-a' 

a^  ' 

2x+2a 

x^-\-2 

ax  +- 

Sx 

4m2- 

-1 

m' 

-f  4m 

2m  +  l 


FRACTIONS 


17T 


12. 


13. 


14. 


15. 


16. 


17. 


18. 


19. 


20. 


21. 


22. 


23. 


24. 


25. 


p2  4-  4  p  -  45  ^  Spr-Sr" 
4:  pr  —  20  r 


p2-81 
a*-Sa^  +  12a^        a"  -  36 


a.'^  +  S 

o^-\-  2x^-^x 

xif-,f 

x"- 

xy-2  if 

a^  +  :ry 

x^- 

2xy-\-y' 

2  a^  +  5  « 

2  +  2 

2  a^  -  3  a 

2  a^  +  a  -  6       6  a^  +  3  a'^ 
6  r^  -  r8  -  2  s2        8  /-^^  -  6  rs  H-  s^ 


12  7^  +  5  rs  -  2  s2 

4r^- 

s^ 

5m4-10      3m  —  9 

8m2- 

-2 

8  m  -  4       Vdm^-h 

3  m^  - 

12 

x^  —  y^          3?  -\-xy  - 

-2.V^ 

5  a;  +  5  y 


2  a^  +  2  2/^  a^  +  3  a.-?/  +  2  2/^      x'-2xy  -\-y' 

8  a?^  -  27  ^  4  ar^  +  12  a;  +  9    ^        4a;-6 

4a^-9    *  8a^  +  12a;4-18  '  6a;2_}_53._6* 

m"*  —  1  4  m^  —  3  »i     4  m^  +  3  w 


16  m^  -  9  n2       2  m^  -f  2 
2a^-a     4a2  +  4a  +  l 


4a2-l 
1-ar^ 


5a 
ar^-4 


m  —  1 

10  g^  -  10  g 
4g2_2g-2 

3  a) -6 


2aj-4      ar^_a;-2     ar^  +  a;  -  2 


aP 


^ 


5x 


6a-\-6x 


g2  +  2  ga;  +  ar*     3  x-  3  a     5  a^  +  5  aa;  +  5  x^ 

m^  -f  ^^^  .  n  —  m      [(m  -f  ^Q^  —  'mn'] 
rn?  —  n^     n  -\-  m     [(m  —  nif  +  mw] 


178  ALGEBRA 


DIVISION  OF  FRACTIONS 

135.    In  arithmetic,  to  divide  a  fraction  by  another  fraction, 
we  multiply  the  dividend  by  the  inverted  divisor.     Thus : 

(a)      1.^3_^^_3^ 

^"^^     io-5-;0''3"2 

2 

^  ^     ^        ^       3        5       3       Z;      3 

In  algebra,  it  is  advisable  to  factor  the  expressions  first,  and 
then  use  the  same  process  as  in  arithmetic. 

Example.     Divide  ^-11^  by  ^-^^  +  1. 
Solution:!,  o.^^  ^  ?iziM±i 

2  a;(x-l)  .  (x^l)(x-l) 
(x  +  l)(ar.2-a;  +  l)  ■    a;(x2-a;+l) 

1  1 

3  x(o^^^)  x(x2.^='-x-^fT:) 

(X  +  1)(^!^=^TT)      (^^-l)(iK  -  1) 
1  1 

x^  x^ 


(x  +  l)(a;-l)      x^-l 
Check  :  Let  x  =  2. 

x2  -  a;  ^  2     x2  -  2  X  +  1  ^  1    ^^^^  2  ^1^1. 
a^s+l      Q'    ofi^x^  +  x       6'  9      6      3' 

also,  ^ —  =  -  • 

'  a:2  -  1      3 

Rule.  —  To  divide  one  fraction  by  another : 

1.  Factor  the  numerators  and  denominators  of  the  fractions. 

2.  Invert  the  divisor  fraction. 

3.  Multiply  the  dividend  fraction  by  the  inverted  divisor. 


FRACTIONS  179 


EXERCISE  83 
Perform  the  following  indicated  divisions : 

5. 


7. 


5    .  10 
18  •   9  ' 

3     21c2  .    3c 
'    32d  '  Sd^ 

7a   .  5a 
15b     66 

15ar^  .  Sx" 

3a         . 

2?               10 

aj2_6a;-|-8  * 

ar^ 

-xll2         '''• 

4r-'-25?r^  . 

,_2_ 

rc-5^^        11. 

32  m*' 


15  n^      35  w8 


_  II*  tf  U/  «  -1-tf  a/   u  (^  .(/  ^4  at)  /^       91.9 

2.     ^z;^  -^  rrr  •  4.     — — ^  H-  t— „  •  6.     —-7-^  -^  6  a26l 

OX*jf 

(c-rf)-  '    (c  +  d)  ' 

16r2-9y2  '  49-2 -4-3n;         ""'    r^  +  rs-j-s^  *  4r-f  45* 
e'-e    .         (e-iy  ^^     (t-^2sy  .  <8-K2.s^ 


13. 


14. 


15. 


a^m  H-  10  am  H-  21  y«      a^m^  —  9  m^ 


tv^  —  H         ,  iv'  +  2ic-{-A 
w^-\-7w-^10  '      w^-\-2w 

a^-5ab-Ub^     a^-3a6-286' 
a^-f  5a6-2462  *  a2-8a64-156«' 


m^  —  y^    .  ^^  4-  ^^/  H-  ^?/ 
mY  —  y*  '         mif-irf 

Perform  the  indicated  multiplications  and  divisions 
^^     x'  +  7xy  +  10tf  ^  x  +  y 


18. 
19. 
20. 


x^ -\- Q  xy  -\- 5  y^      op^ -\- 4  xy -\- 4:  y^     x-\-2y 

a:'-b^         ^  ab-2b\   (a-  bf 
a2-3tt64-26=*  *    62  +  06    'a{a-b)' 

r2-|-4rs-|-4.s2  *  4r^ -^ ^^ hr"^ ^lOrs 

am*  —  an*  _^  m^  +  n^       bm^  -\-  bmn  -\-  bn^ 
bm^  —  bn^     m^  —  n-     aw?  +  2  amn  +  ai^ 


180 


ALGEBRA 


136.  Sometimes  there  are  mixed  expressions  in  an  algebra 
problem.  These  mixed  expressions  should  always  be  reduced 
to  fractional  form  as  in  arithmetic. 


Example  1.     Simplify  12f  -^  3\. 
Solution  : 


123 -.31=^-.  1^  =  ^x1  =  ^. 
*        *       8        4       8       13     26 


Example  2.     Simplify  (5 


a' -19:. 


4:X^ 


^ )     \        a-2xj 


Solution  :  In  the  first  parentheses,  the  fraction  is  to  be  subtracted 
from  5,  and  in  the  second,  the  fraction  is  to  be  subtracted  from  3.  (Use 
rule,  §  133.) 


■  V         a2-4xV  ■  V        a -2x1 

/5(a2-4x2)-((f2_i9a;2)\  ^  /3(a  -  2a;)-(a  -  5x)\ 
\  a^-4x^  )  '  \  a-  2x  J 

/5a2-20a;^-a-2  +  19x^\  _^  /Sa-6x-  a-]-6x\ 
[  a^-^x^  j  '  \  a-2x  J 


W-Ax-^l  ■  [a- 


2x 


{a^^^){a  +  2x)      (2,0--^      a  +  2x 


Check  :  Let  a  =  1 ;  x  =  1. 
g_a^-]9^^5_l-a9^5 


18 


a--2-4  x2 


1-4 


=  6-6: 


3-^~^^  =  3-J^^  =  3-  — =3-4=-l, 


also, 


a-2x  1-2 

2aH-x^2  +  1^3^.^ 
a-{-2x     1  +  2     3 


and 
(-l)H-(-l)=l; 


Note.  In  such  examples,  first  perform  carefully  the  indicated  additions 
and  subtractions  loithin  the  parentheses  and,  afterwards,  the  multiplications 
and  divisions. 


FRACTIONS  181 

EXERCISIi:  84 
Perform  the  indicated  operations : 

1.  (3+t).(3  +  A)..  7.    (2-|)(J^,). 

2.  (5-i).(2-T%).  fa'      a\^f2a'     a\ 

V     yJ   vf-^J  9, 


...  (.,._^),(._.-?). 

/2m'^  +  7m-15\    /2  m'^  -  19  m  +  42\  .  A 

■  [2m'-3m-Uj'\        8m-12        y\ 

V^-jr     -ua     ary  \       a;  -  vj      \v     xj 

20    f    ^^^'^    ^  '    ra;'^-3a;  +  9     x'  +  x-6]_ 

■  \x'  +  x-12j'   [x'-^-'^x-S'    3a;-9    J* 


3m+6 


182  ALGEBRA 


COMPLEX  FRACTIONS 

137.  A  Complex  Fraction  is  a  fraction  having  one  or  more 
fractions  in  either  or  both  of  its  terms ;  as, 

(^^ «_\ 

d  \a  —  b     a  +  bj 

138.  A  complex  fraction  is  simply  a  case  of  division  of 
fractions.  The  problems  are  very  similar  to  those  in  the  last 
paragraph.  The  numerator  and  denominator  should  be  sim- 
plified separately  and  then  the  division  performed. 

Example  1.     Simplify 


d 


Solution  :  This  means  divide  a  by  the  result  obtained  by  subtracting 

-  from  6. 
d 

ad 


a     _        a        _       I     ^     \ 
_c~  /bd-c\~^'  \bd-c) 
d      \     d     J 


bd 


Check  :  Let  a=:l,  6  =  2,  c  =  l,d  =  l. 
g     _     1     _     1     ^1 

d  1 

also,  «^  ^     -^ 


=  1; 


bd-c      2 


/    a a    \ 

Example  2.     Simplify  ^ — 7 r 


This  means  :  find  the  difference  of  the  fractions  in  the  numerator,  and 
the  sum  of  the  fractions  in  the  denominator ;  divide  the  first  result  by  the 
second.     The  work  may  be  arranged  as  follows  : 


FRACTIONS  183 


Solution  :  1.      — ; 


4.    = 


a  —  h     a  -\-  b 
2     _a(a-\-b)—  a(a-b)  , -bCa  +  h)+ a(a- b) 

Ca-6)(a  +  &)  (a-6)(a  +  6) 

o     _  a^-^ab  —  a^-^  ab  ^  ab  +  b'^ -h  a^  -  ab 
ia-b)ia  +  b)     '    (a-6)(a+6) 
2  a6  .  ga  +  ?>^ 

(a-?v)(a  +  ?>)  ■  ia-b)(a  +  b) 
5   _  2  aft  ^(a-b)(a  +b)  _    2ab 

(a  -  6)(a  +  6)  a^  +  b'^  a^  +  b'^ ' 

Check  :  Let  a  =  2  ;  6  =  1. 

a  a         2_2     4 

a  -  &     a  +  6      1      3_3_4      3_4 


J) ■       a         1,2      6     3      5      5 

:-b     a+b      13     3 

*  a2  +  62     4  ^.  1      5 


EXERCISE  85 

«^^  e.  ^.  10.  -^. 


2.   i^t_SL. 


7.        '' 


ar'-v' 


1-i 


?•  6  11- 

8.        3 


4  1+^ 

6-*  2 

771 


12. 


a- 

1 
'4a 

P- 

1 

1  - 

1 

1> 

1 

-  X 

1  —  ^  m  or 


184  ALGEBRA 


13.   ^^.  16.   ^.  17.    -^-JL 


5      d                                           a;  a  +  2  6 

14.   ^-^^      „^".        16.   ^.         18.       ^       "  • 


2/     a?  X  y  X 

,2^ 


n       m  —  n  9  x^  —  y^ 

n      m-\-n  '   -.      2(x+2  y) 


m 

0?- 

m  —  n 

-^ 

6  + 


3a.' +2/ 
17  a;  +  22 


a^-3i»-4 
20. : 23.    5 

2+     ^ 


a 


X   —    4: 

a  —  x 


21.       "  +  f     .  .  24.  \+  "^  ■ 

■1,0  H       g^  — go? 

a  —  6  l-\-ax 

2a— h  .  2a+h 


25. 


a4-3&  CT-3& 
2«-6  2ct  +  6' 
a  — 36     a  +  3  h 


XL   SIMPLE  EQUATIONS— (Conf/ni/eJ) 

FRACTIONAL  EQUATIONS 

139.  If  the  unknown  number,  or  numbers,  of  an  equation  do 
not  appear  in  the  denominator  of  a  fraction,  the  equation  is 
called  an  Integral  Equation  ;  as,  3  a;  —  5  =  2  a;  -f-  7. 

140.  If  the  unknown  number,  or  numbers,  of  an  equation  do 

appear  in  the  denominator  of  a  fraction,  the  equation  is  called 

3  2 

a  Fractional  Equation ;  as, 5  =  -  +  7. 

X  X 

141.  A  fractional  equation  may  be  changed  into  an  integral 
equation  by  Clearing  of  Fractions. 

Example.     Solve  the  equation  — — ^-  =  4h — -^ — 

^  4  5  10 

Solution  :  1.  The  lowest  common  multiple  of  4,  5,  and  10  is  20.  Mul- 
tiply both  members  of  the  equation  by  20. 

2.  /p  .  ('^^-^)-/p  .  (i^^^  =  20  .  4  +  ;|.  ^I^±il. 

3.  .-.  6(3x- l)-4(4x-6)  =80  +  2(7ar  +  6). 

4.  .-.  loa;-5  -  16j; +  20  =  80  +  14a:  + 10. 
6.  .-.  15  -  a;  =  90  +  14  x. 

6.  /.  -  15  a;  =  76. 

7.  D_i5:  x=-5. 

Check:  Does  3(- 5)- 1  _4(- 5W  5  ^         7(- 5)+ 5  ^ 
4  5  10 

Does   ml^  _  ^J^  =  4  4- ZL?P  ?      Does -4 +  5  =  4- 3?    Yes. 
4  5  10 

18ft 


186  ALGEBRA 

EXERCISE  86 
Solve  the  following  equations  : 
1     a^4-5     a^-fl_3  ^    ?r -f- 12  _  ?r  —  9 


2  4 

10 


2.    -^"^  +  2  =  •^^"^^-  8.    "'^  "^  ""  —  ""^  ~  '"^  =  —  1. 


11  +  m,      10  —  m  _  ^ 


9 

2 

7?l  +  ll 

10-??i 

6 

3 

^-1, 

f-M4 

6  3  3  9 


4.  l(a  +  5)-4  =  ^^^^.  10.    1+1?_5(._3)=4. 
3^       ■                    4  4  7^^ 

5.  K^  +  2)-i(r-2)=2.       11.    ^^{Uu+l)-\(i^-^)  =  l. 

c-2     c-4^2  J2    3(.T-1)  ■    5a?H-7^17/ 

4  6         3*  *         lo"  3  6* 

13.  il±i_l(2Z>+3)=^i^. 

2  o  4 

14.  ^-|(4n4-9)=l(5n  +  8). 

15.  ^^-^(8^  +  3)  =  l(4^-3). 

7a;  — 8      7  x-\-6  _x  —  r)     4a;-f9 
14  4  a;     "     2  7  a;    * 

Solution:  1.    Multiply  both  membei's  by  28a:,  the  L.  C.  M.  of  the 
denominators. 

8.  .-.  2  a-C7  X  -  8)  -  7(7  x  +  6)  =  14  x(x  -  5)-  4(4  x  +  9). 

4.  /.  14  a;2  _  16  x  -  40  x  -  42  =  14  x^  -  70  x  -  16  x  -  36. 

6.  ...  _65x-42  =-86x-36. 
Q.  .•..+  21x=4  6. 

7  .     /,.  _     6     _    2 

Check  the  solution  either  by  substitution  or  by  going  over  the  steps 
carefully. 


SIMPLE   EQUATIONS  187 

It  is  essential  that  roots  of  fractional  equations  be  checked. 
For  reasons  that  are  given  in  a  later  course  in  algebra,  whenever 
the  apparent  root  makes  the  denominator  of  any  fraction  zero, 
that  number  is  not  a  root  at  all. 

n.  6_U5.  22. 

a     a     o 

18.    A_A=  7.  23_ 


6_ 

a 

1      5 
a      3* 

5 
4:m 

2 
3  m 

7 
48* 

3 
4a; 

+  ^  =  15 

X 

h 

X  - 

■2     4 

2. 

5  X            X 

12 

X 

2x-\-S 
5x 

+  3 

=  0. 

5^4-4 

ll«-2_3 

2i 

■    6^      -•"• 

3m -5 
4 

9  7^1  -  7        2 
12          3  m 

19.    ^  +  ^  =  151.  24.   30_1^9+   7  ^0_ 

7  21  a;  3x 

5x-4.     10a;  +  9^51 
5  10  6  a;' 

r_3      1     3r-7 


20.      r=-_2.  25. 


21.   ±r_!l^^_^+3=0.  26. 


2  r       3         2  r 

2  5  2 

27.  Solve  the  equation =  0. 

1  a;-2a;  +  2a;2_4 

Solution  :  1.   The  L.  C.  M.  of  (x-2),  (x+2),  and  (x^  -  4)  is  (a;2-4). 
Multiplying  both  members  of  the  equation  by  x^  —  4, 

3.  .-.  2{x  +  2)  -  5(x  -  2)  -  2  =  0. 

4.  .-.  2  ic  +  4  -  5  a;  +  10  -  2  =  0. 

5.  .-.  -3a;  +  12  =  0. 

G.  .♦.  -  3  X  =  -  12,  or,  a;  =  4. 

28.  -12.      3.  32.   l^  =  -i2_ 
x-2  1-x     d-x 

29.  .^0_  =  _9_.  33.      2x    _4x+5        3 


771  —  3     m  — 5  3x— 4     6a;— 1     3a;— 4 

30.  i±5  +  _l_  =  5.  34.    Igj^'-Sa^-S^g 
x-3      a;-3  3x»  +  6a;  +  4 

31.  5^  +  1^3^      X  _        35      6»  +  5  ^    2     _   3a: 


2a;-3  2a:-3  2ar'-2a!    x'-l      af^-l 


188  ALGEBRA 

3m  2m         2m^-15 


36. 
37. 
38. 
39. 


2m  +  3      2m-3      4m^ 

9 2^1 

3^-5      t-2     t-3 

3  4  1 


,._2     2r-l      r  +  4 

2(y-7)  y-2     ?/  +  3^q 


40    a  +  2      2a-3_     26  -  a 


41. 


42. 


43. 


a  —  4       a  +  3       a^  —  a  — 12 

3w 4        ^1 

2m-6     5m- 15     2* 

3a  7      ^3 

2a-5     3a  +  12* 

2:24-1        2^-1  9^4-17 


44. 


45. 


22-16      2:^4-12      z'-2z-^S 

2t  +  7     3t  —  5^17t  +  7 
6f-4     9^4-6      9«2_4' 

3a;-2^36-4a;     24-3a; 
a;4-3-       x'  —  9         S  —  x 

,'       o  1      ^v  4.-       6x4-1       2a;-4       2aj-l 

46.    Solve  the  equation  — — ^ -—  =  — 

^  15         7  a; -16  5 

Solution:  1.    Clear  of  fractions  only  partially  at  first  by  multiplying 
by  the  L.  C.  M.  of  15  and  5. 

Mis:  6a^4-l-^y^~^^  =  6x-3. 

1  X  —  lb 

QQ  /v.  go 

2.  Transposing  and  uniting  terms,      4  =  - —- • 

7  x  —  16 

3.  M(7  x-16) :  28  a:  -  64  =  30  a;  -  60. 

4.  Completing  the  solution,  x  =  ~2. 
Check  it  by  substitution. 


SIMPLE  EQUATIONS  189 


47. 
48. 
49. 


2a-l 

a+2 

6a-5 

2 

2a  +  5 

6 

y+  ^y 

3^33/- 

'     _  9  ?/  - 
-4:          9 

-2 

2a;  4-7 

5a;-4 

JB-f  6 

14 

3«-hl 

7 

ot  +  1 

4t-\-7 

3f-2 

5  5«  +  8  3 


EXERCISE  87 

1.  Divide  56  into  two  parts  such  that  five  eighths  of  the 
greater  shall  exceed  seven  twelfths  of  the  less  by  6. 

2.  If  the  base  of  a  certain  rectangle  be  increased  by  2  feet, 
the  altitude  is  equal  to  one  third  of  tlie  result.  The  perimeter 
of  the  rectangle  is  36  feet.  Find  the  base,  altitude,  and  area 
of  the  rectangle. 

3.  A  has  $52  and  B  has  $38.  After  giving  B  a  certain 
sum,  A  has  left  only  three  sevenths  as  much  money  as  B  then 
has.     AVhat  sum  was  given  to  B  ? 

4.  Divide  45  into  two  parts  such  that  the  sum  of  four 
ninths  of  the  greater  and  two  thirds  of  the  smaller  shall 
equal  24. 

5.  The  denominator  of  a  certain  fraction  exceeds  the 
numerator  by  27 ;  if  9  be  subtracted  from  both  terms  of  the 
fraction,  the  value  of  the  fraction  becomes  \.  Find  the 
fraction. 

6.  A's  age  is  three  eighths  of  B's,  and  8  years  ago  it  was 
two  sevenths  of  B's  age.     Find  their  ages  at  present. 

7.  Washington  was  admitted  to  the  Union  18  years  before 
Oklahoma,  and  may  therefore  be  said  to  be  18  years  older 
than  Oklahoma  as  a  state.  One  fourth  of  Washington's  age 
in  1911  exceeded  Oklahoma's  age  by  1^  years.  Find  the 
year  when  each  was  admitted.' 


190  ALGEBRA 

8.  If  a  certain  number  be  diminished  by  23,  one  fourth  of 
the  result  is  as  much  less  than  37  as  the  number  is  greater 
than  56.     Find  the  number. 

9,  If  the  number  of  states  admitted  to  the  Union  since  its 
formation  by  the  13  original  states  is  diminished  by  9,  the 
quotient  obtained  by  dividing  that  number  by  2  equals  the 
original  number  of  states.  Find  the  present  number  of  states 
(1912). 

10.  The  numerator  of  a  certain  fraction  is  6  less  than  the 
denominator;  if  the  denominator  is  increased  by  1  and  the 
resulting  fraction  be  multiplied  by  3,  the  product  equals  |. 
Find  the  fraction. 

11.  Find  the  angle  such  that  3  times  its  complement  in- 
creased by  two  thirds  of  its  supplement  equals  137°. 

12.  I  buy  some  bulbs  from  a  seed  store  for  $3,  paying  75  p 
per  dozen  for  one  variety,  and  50^  per  dozen  for-  another 
variety.  The  number  of  the  first  variety  purchased  exceeds 
the  number  of  the  second  variety  by  18.  Find  the  number 
of  each  variety  purchased. 

13.  If  a  railroad  train  consists  of  a  certain  type  of  passen- 
ger engine,  one  parlor  car,  and  five  sleeping  cars,  its  value  is 
$  129,200.  The  value  of  each  sleeping  car  exceeds  the  value 
of  the  engine  by  $300;  the  value  of  the  parlor  car  is  five 
sixths  of  the  remainder  when  the  cost  of  the  engine  is  dimin- 
ished by  $  100.  Find  the  value  of  the  engine,  the  parlor  car, 
and  of  a  sleeping  car. 

14.  A  man  has  $3000  invested,  part  at  5  %  and  part  at  6  %. 
His  total  income  per  year  is  $157.  How  much  has  he  in- 
vested at  each  rate? 

15.  In  1912,  the  "age"  of  Maine  was  4y2-j-  times  that  of 
Wyoming ;  in  1920,  it  will  be  3i  times  it.  Find  when  each 
state  was  admitted  to  the  Union. 


SIMPLE  EQUATIONS  191 

16.  The  denominator  of  a  certain  fraction  is  7  less  than  the 
numerator ;  if  5  be  added  to  the  numerator,  the  value  of  the 
fraction  becomes  f .     Find  the  fraction. 

17.  The  income  at  5%  on  one  sum  of  money  exceeds  by 
$  35.50  the  income  at  4  %  on  a  sum  which  is  $  350  Jess  than 
the  first.     Find  the  two  sums  invested. 

18.  The  denominator  of  a  certain  fraction  exceeds  the 
numerator  by  5 ;  if  the  denominator  be  decreased  by  20,  then 
the  resulting  fraction,  increased  by  1,  is  equal  to  twice  the 
original  fraction.     Find  the  fraction. 

19.  The  supplement  of  a  certain  angle  divided  by  its  com- 
plement gives  as  quotient  2^.     Find  the  angle. 

20.  If  twice  a  certain  number  be  diminished  by  5  and  the 
result  be  divided  by  the  number,  the  quotient  exceeds  1  by  a* 
fraction  whose  numerator  is  7  more  than  the  number  and 
whose  denominator  is  3  less  than  the  number.  Find  the 
number. 

142.  Work  Problems.  If  a  man  can  do  a  piece  of  work  in 
8  days,  then  in  one  day  he  can  do  one  eighth  of  it,  and  in  three 
days  he  can  do  thi'ee  eighths  of  it. 

If  a  man  can  do  a  piece  of  work  in  x  days,  then  in  one  day 

1  15 

he  can  do  -  part  of  it,  and  in  5  days  he  can  do  5  x  -  or  - 

X  XX 

part  of  it. 

EXERCISE  88 

1.  If  a  man  can  plow  a  field  in  15  days,  what  part  can  he 
plow  in  one  day?  in  4  days?  in  x  days? 

2.  If  a  machine  can  do  a  piece  of  work  in  x  days,  how 
much  can  it  do  in  one  day  ?  in  7  days  ? 

3.  A  can  do  a  piece  of  work  in  5  days,  and  B  the  same 
work  in  8  days. 

(a)    How  much  can  A  do  in  one  day?  in  x  days? 
(6)  How  much  can  B  do  in  one  day  ?  in  a;  days  ? 


192  ALGEBRA 

(c)  How  much  can  they  do  together  in  one  day  ?  in  a;  days  ? 

(d)  How  much  can  A  do  in  2  days  ? 

(e)  How  much  can  B  do  in  3  days  ? 

(/)  How  much  can  they  do  together  if  A  works  2  days  and 
B  3  days? 

Equations 

4.    A  can  do  a  piece  of  work  in  10  hours  and  B  can  do  it  in 

5  hours ;  how  long  will  it  take  them  to  do  it  together? 

Solution  :  1.  Let  x  =  the  number  of  hours  it  will  take  them  to  do  it 
together. 

2.  A  does  4t  in  1  hour  ;  .-.  he  will  do  —  in  aj  hours. 

B  does  ^  in  1  hour  ;  .-.  he  will  do  -  in  x  hours. 

5 

.-.  they  will  do  (  —  +  -  )  in  a;  hours. 

VlO    5y 

3.  They  complete  the  task  in  x  hours.  Represent  the  whole  task  by 
^  of  itself  or  by  1. 

4.  .-.  -^  +  ^  =  1. 

10     5 

5.  Mio  :  a:  +  2  a;  =  10. 

6.  3  a;  =  10,  a:  =  3^  hours,  or  3  hours  and  20  minutes. 

5.  A  painter  can  paint  a  house  alone  in  5  days,  and  an 
apprentice  can  do  it  alone  in  15  days.  In  how  many  days  can 
they  do  it  if  they  work  together? 

6.  A  man  can  plow  a  certain  field  in  6  days;  his  son  can 
plow  it  in  9  days.  How  long  will  it  take  them  to  plow  the 
field  if  they  work  together  ? 

7.  In  a  newspaper  office  there  is  one  machine  which  can 
print  the  morning  issue  of  the  paper  in  2  hours,  and  another 
which  can  do  it  in  3  hours.  In  how  many  hours  can  they  turn 
out  the  edition,  if  they  are  run  together? 

8.  A  can  do  a  piece  of  work  in  15  hours,  and  B  can  do  it 
in  18  hours.  If  A  works  for  7  hours,  how  many  hours  must 
B  work  to  complete  the  task? 


SIMPLE   EQUATIONS  193 

9.  A  can  do  a  piece  of  work  in  15  hours,  while  B  can  do 
it  in  25  hours.  After  A  has  worked  a  certain  time,  B  com- 
pletes the  work.  If  B  works  9  hours  longer  than  A,  how  long 
did  A  work? 

10.  A  can  do  a  piece  of  work  in  18  days.  If  he  and  B  can 
do  three  fifths  of  it  in  6  days,  how  long  will  it  take  B  alone  to 
do  the  work  ? 

143,  Problems  about  the  Lever.  A  teeter  board  is  one  form 
of  lever.  The  point  which  supports  the  lever  is  called  the  ful- 
crum; the  parts  of  the  lever  to  the  right  and  left  of  the  fulcrum 
are  called  the  lever  arms. 

If  the  weights  L  and  R  just  balance,  it  is  well  known  that, 
if  R  moves  farther  to  the  right,  while  L  is  stationary,  then 
the  right  side  goes  down ;    and  if 
R  moves  toward  the  fulcrum,  then      h  ^ 

the  right  side  goes  up.     Thus,  the 

influence  of  R  upon  the  lever  depends  both  upon  the  weight  of 
R  and  its  distance  from  the  fulcrum. 

The  influence  of  a  weight  upon  a  lever  is  called  its  leverage. 
It  can  be  shown  that  the  leverage  of  a  weight  is  measured  by  the 
product  of  the  weight  and  its  distance  from  the  fulcrum. 

Thus,  if  R  weighs  50  pounds  and  is  4  feet  from  the  fulcrum,  its  lever- 
age is  200  ;  and,  if  L  weighs  80  pounds  and  is  2^  feet  from  the  fulcrum, 
its  leverage  is  80  x  2^  or  200  also.     The  two  wiU  balance. 

This  truth  may  be  tested  in  the  following  manner. 
1.    Remove  a  side  and  an  end  from  a  crayon  box.     Balance  a  stiff  ruler 
on  the  edge  of  the  box  (the  fulcrum). 

2.    Place  two  pennies  6  inches  from  the  ful- 
crum on  the  left  side.     Find  where  four  pen- 
nies must  be  placed  on  the  right  side  of  the 
fulcrum  to  balance  them.    (It  should  be  3  inches 
to   the   right.)      Notice   that    3x4=12   and 
that  6x2  =  12,-    that  the  leverages  are  equal. 
3.   Find  where  3  pennies  must  be  placed,  on-  the  right,  to  balance  the 
2  pennies  on  the  left.     Find  the  leverage  of  the  3  pennies  and  compare 
it  with  the  leverages  in  step  2. 


194  ALGEBRA 

4.  Keeping  the  two  pennies  C  inches  to  the  left,  place  two  others 
2  inches  to  the  right,  and  2  four  inches  to  the  right.  The  ruler  should  be 
in  perfect  balance  again. 

The  leverage  of  the  first  two  pennies  on  the  right  is  2  x  2  or  4  ;  of  the 
second  two,  is  2  x  4  or  8  ;  4  +  8  —  12. 

Example.  Suppose  that  the  weights  and  distances  in  the 
figure  are : 

A.  40  pounds  ;  distance  5  feet. 

B.  45  pounds  ;  distance  4  feet. 

C.  55  pounds ;  distance  6  feet. 

D.  60  pounds ;  distance  6  feet. 

E.  50  pounds ;  distance  x  feet. 

Where  must  E  be  placed  so  that  the  lever  will  balance  ? 

CAB  D  E 


Solution  :  1.  The  leverages  are  : 

A.  6  X  40  =  200.  C.   6  X  55  =  330.        J?.   50  x  a;  =  50  x. 

B.  4  X  45  =  180.  D.   6  x  60  =  360. 

A  B  C  D  U 

2.  .  •.  200     +     180     +     330     =     360     +     50  x. 

.-.350  =  50  a; 

.-.      7  =  x. 

Rule.  —  To  make  a  simple  lever  balance : 

The  sum  of  the  leverages  of  all  weights  (forces)  on  one  side  of  the 
fulcrum  must  equal  the  sum  of  the  leverages  of  all  weights  (forces) 
on  the  other  side  of  the  fulcrum. 

EXERCISE  89 

1.  A  boy  weighing  70  pounds  sits  6  feet  from  the  fulcrum 
and  balances  a  boy  who  is  sitting  3^  feet  from  the  fulcrum  on 
the  other  side.     Find  the  weight  of  the  second  boy. 

2.  A,  weighing  96  pounds,  sits  5^  feet  to  the  left  of  the 
fulcrum.  If  B  weighs  66  pounds,  where  must  he  sit  on  the 
right  in  order  to  balance  A  ? 


SIMPLE   EQUATIONS  195 

3.  A,  who  weighs  92  pounds,  and  B,  who  weighs  115  pounds, 
wish  to  sit  at  the  ends  of  a  teeter  board  wliich  is  9  feet  long. 
How  far  from  A  must  the  fulcrum  be  placed  so  that  they  will 
balance  ? 

4.  A  boy  wishes  to  carry  two  heavy  packages  over  his 
shoulder  by  balancing  them  at  the  ends  of  a  stiff  rod  which  is 
4  feet  long.  If  one  package  weighs  20  pounds  and  the  other 
80  pounds,  how  far  from  the  end  upon  which  the  20  pound 
package  is  carried  must  the  rod  rest  upon  his  shoulder  ? 

5.  Three  children,  weighing  G2,  75,  and  89  pounds  respec- 
tively, arrange  themselves  upon  a  teeter  board,  the  first  sitting 
4  feet  from  the  fulcrum,  and  the  second  5  feet  from  the  ful- 
crum on  the  same  side.  Where  must  the  third  sit  in  order  to 
balance  the  other  two  ? 

6.  Three  boys,  A  weighing  73  pound;,  B  weighing  95 
pounds,  and  C  weighing  6o  pounds,  sit  on  a  teeter.  A  is  5  feet 
from  the  fulcrum  on  the  left  side,  B  is  on  the  other  side  4  feet 
away,  C  is  on  the  left  side  4  feet  away.  Can  two  other  boys 
weighing  80  pounds  and  115  pounds  respectively  arrange  them- 
selves one  on  each  side  and  at  equal  distances  from  the  ful- 
crum so  as  to  balance  the  teeter  ?  Where  must  the  boy 
weighing  80  pounds  sit  ? 

144.   Additional  Distance,  Rate,  and  Time  Problems. 

EXERCISE  90 

1.  What  is  meant  by  "the  rate"?  "the  time"?  "the  dis- 
tance "  ?     (§83.) 

2.  Give  a  simple  arithmetical  problem  involving  time,  rate, 
and  distance. 

3.  What  is  the  rule  for  finding  the 

(a)  distance  when  the  rate  and  time  are  known  ? 
(6)  rate  when  the  time  and  distance  are  known  ? 
(c)  time  when  the  rate  and  distance  are  known? 


196  ALGEBRA 

4.  The  rate  of  one  train  exceeds  that  of  another  by  5  miles 
per  hour.     Let  r  represent  the  rate  of  the  slow  train. 

(a)  Express  the  rate  of  the  faster  train. 
(p)  Express  the  time  required  by  each  train  in  going  100 
miles. 

5.  The  rate  of  one  train  is  f  that  of  another.  Let  x  repre- 
sent the  rate  of  the  faster  train. 

(a)  Express  the  time  each  requires  for  a  trip  of  50  miles. 
(6)  Form  an  equation  to  express  the  fact  that  the  time  of 
the  slow  train  exceeds  that  of  the  faster  train  by  1  hour. 

6.  The  time  required  by  one  train  in  going  a  certain  dis- 
tance is  f  that  of  another  train.  Express  the  time  of  the  slow 
train  by  t. 

(a)  Express  the  rate  at  which  each  train  travels  in  going 
a  distance  of  100  miles  in  the  time  mentioned. 

(b)  Form  an  equation  to  express  the  fact  that  the  rate  of 
the  faster  train  exceeds  the  rate  of  the  slow  train  by  20  miles 
per  hour. 

Equations 

7.  The  rate  of  an  express  train  is  three  times  that  of  a 
slow  train.  It  covers  180  miles  in  8  hours  less  time  than  the 
slow  train.     Find  the  rate  of  each  train. 

8.  A  messenger  starts  out  to  deliver  a  package  to  a  point 
24  miles  distant,  at  the  rate  of  8  miles  per  hour.  At  what 
rate  must  a  second  messenger  travel  to  arrive  at  the  same  time 
as  the  first  messenger,  if  he  starts  1  hour  after  him  ? 

9.  The  rate  of  a  passenger  train  exceeds  twice  the  rate  of 
a  freight  train  by  5  miles  per  hour.  It  can  go  350  miles 
while  the  freight  train  goes  150  miles.  Find  the  rate  of  each 
train. 

10.  A  man  just  missed  a  train.  He  knew  that  it  would 
stop  at  a  station  15  miles  distant  from  the  central  station  and 


SIMPLE   EQUATIONS  197 

decided  to  try  to  catch  it  by  going  to  the  second  station  in  an 
automobile.  If  the  train  runs  at  the  rate  of  20  miles  per  hour, 
at  what  rate  must  he  travel  in  the  automobile  in  order  to 
arrive  at  the  station  10  minutes  ahead  of  the  train,  if  it  takes 
him  5  minutes  to  get  the  automobile  and  if  it  is  20  miles  to 
the  station  by  road  ? 

River  Problems 

On  a  river,  the  direction  in  which  the  water  is  flowing  is  called  down- 
stream,  and  the  opposite  direction  is  called  upstream.  When  going 
downstream,  a  boat  is  carried  along  by  the  current  of  the  river  and  what- 
ever force  is  exerted  within  the  boat ;  when  coming  upstream,  its  progress 
is  retarded  by  the  current  of  the  river.  This  is  something  like  the  effect 
of  the  wind  upon  a  person  who  is  walking  ;  when  going  with  the  wind,  he 
is  carried  along  by  it,  and  when  going  against  the  wind,  he  is  retarded 
by  it. 

11.  The  rate  of  the  current  of  a  river  is  3  miles  per  hour: 
(a)  at  what  rate  will  some  boys  go  downstream  if  their  own 
rowing  is '  at  the  rate  of  5  miles  per  hour  in  still  water  ? 
(h)  upstream  ? 

12.  How  long  will  it  take  the  boys  in  Example  11 :  (a)  to 
go  24  miles  downstream  ?  (b)  24  miles  upstream  ?  (c)  for 
the  trip  down  and  back  ? 

13.  (a)  How  long  will  it  take  the  same  boys  to  go  d  miles 
downstream  ?  (h)  d  miles  upstream  ?  (c)  Form  an  equation 
to  express  the  fact  that  the  time  down  and  back  is  5  hours, 
(d)  Find  d  from  the  equation  in  step  (c). 

14.  Some  boys  who  can  row  4  miles  an  hour  in  still  water 
made  a  trip  on  a  river  whose  current  is  2  miles  an  hour.  If  it 
took  them  8  hours  for  the  trip,  how  far  did  they  go  ? 

15.  A  party  take  a  trip  in  a  motor  boat  which  runs  at  the 
rate  of  15  miles  an  hour.  They  take  3  hours  for  the  trip. 
What  distance  did  they  go,  if  the  rate  of  the  current  is  3  miles 
an  hour  ? 


198  ALGEBRA 

EXERCISE  91 

Supplementary  Problems 

1.  Three  times  the  difference  between  one  fourth  and  one 
tenth  of  a  certain  number  exceeds  five  by  one  fifth  of  the 
number.     Find  the  number. 

2.  A's  age  11  years  from  now  divided  by  his  age  11  years 
ago  is  the  fraction  -i/--     Find  his  present  age. 

3.  The  width  of  a  room  is  three  fifths  of  its  length;  if  12 
feet  be  added  to  the  width  and  taken  from  the  length,  the  room 
will  be  a  square.     Find  its  dimensions. 

4.  Five  lines  radiate  from  a  point  making  angles  such  that 
the  second  is  one  half  of  the  first,  the  third  is  twice  the  first, 
the  fourth  is  the  sum  of  the  second  and  third,  and  the  fifth  is 
three  times  the  third.     Find  the  angles.     (See  §  13.) 

5.  Find  the  three  angles  of  a  triangle  if  the  second  angle 
is  one  half  of  the  remainder  obtained  by  diminishing  the  first 
angle  by  1°,  and  if  the  third  angle  is  |  of  the  remainder  ob- 
tained by  diminishing  the  first  angle  by  7°.     (See  §  13.) 

6.  Two  men,  A  and  B,  57  miles  apart,  travel  towards  each 
other,  B  starting  20  minutes  after  A.  A  travels  at  the  rate  of 
6  miles  an  hour  and  B  at  the  rate  of  5  miles  an  hour.  How 
far  will  each  have  traveled  when  they  meet  ? 

7.  Washington's  Monument,  the  highest  piece  of  masonry 
in  the  world,  consists  of  a  main  shaft  surmounted  by  a  pyra- 
mid. The  height  of  the  main  shaft  exceeds  that  of  the  pyramid 
by  445  feet ;  if  the  height  of  the  pyramid  be  decreased  by  5 
feet  and  the  result  be  divided  by  the  height  of  the  main  shaft, 
the  quotient  is  the  fraction  -^.  Find  the  height  of  the  two 
parts  of  the  monument. 

8.  The  rate  of  an  express  train  is  five  thirds  of  that  of  a 
slow  train.  It  travels  36  miles  in  32  minutes  less  time  than 
the  slow  train.     Find  the  rate  of  each  train. 


SIMPLE   EQUATIONS  199 

9.  If  the  third  of  three  consecutive  even  numbers  is  divided 
by  each  of  the  first  two  in  turn,  the  difference  of  the  fractions 
obtained  is  equal  to  the  quotient  of  7  divided  by  three  times 
the  first  number.     Find  the  numbers. 

10.  A  workman  does  one  third  of  a  piece  of  work  in  5 
days ;  he  and  a  second  workman  complete  the  task  together  in 
4  days.  How  manys  days  would  it  take  the  second  man  to  do 
the  work  alone  ? 

11.  Some  boys  row  on  a  river  whose  current  is  known  to  be 
2^  miles  per  hour.  They  find  that  it  takes  them  as  long  to  go 
upstream  2  miles  as  downstream  7  miles.  What  is  their  rate 
of  rowing  ? 

12.  A's  age  exceeds  twice  B's  age  by  7  years.  The  sum  of 
three  fifths  of  B's  age  2  years  ago  and  of  four  sevenths  of  A's 
age  4  years  from  now  is  26  years.     Find  their  present  ages. 

13.  The  income  on  one  sum  of  money  at  4|^  %  and  the  in- 
come on  a  sum  $  600  greater  at  3^  %  together  amount  to  $  421 
per  year.     Find  the  total  amount  invested. 

Hint:  H%=n  =  'Eh- 

14.  The  rate  of  an  express  train  is  three  halves  that  of  a 
slow  train.  It  covers  270  miles  in  3  hours  less  time  than  the 
slow  train.     Find  the  rate  of  the  train. 

15.  A  water  reservoir  can  be  filled  by  an  old  pump  in  12 
hours.  After  a  new  one  is  installed,  it  is  found  that  the 
reservoir  can  be  filled  by  the  two  pumps  together  in  4  hours. 
How  long  would  it  take  the  new  pump  alone  to  fill  the 
reservoir  ? 

16.  A  man  invests  a  sum  of  money  in  4^%  stock  and  a 
sum  %  180  greater  than  the  first  in  3^  %  stock.  If  the  incomes 
from  the  two  investments  are  equal,  find  the  sums  invested. 

17.  Some  boys  are  rowing  on  a  river  whose  current  is  5 
miles  per  hour  in  one  stretch  and  3  miles  in  another.  They 
find  when  going  downstream  that  they  can  go  4  miles  where 


200  ALGEBRA 

the  current  is  rapid  in  the  same  time  that  they  can  go  3  miles 
where  the  current  is  slower.  Find  the  rate  at  which  they  row 
in  still  water  ? 

The  following  three  equations  arise  in  the  solution  of  three 
problems  in  applied  mathematics.     Solve  them. 

18.  S0T-^gT'=:S0(T-2)^^g(T-2y,wheveg  =  S2.     De- 
termine T. 

19.  ^Jil:!  =  13.6  +  ^^:^ .     Get  the  result  to  two  decimals. 

X  X 

20.  ^^  +  'M^  =  xv.     Determine  a.. 

SOLUTION  OF  LITERAL  EQUATIONS 

145.   A  Literal  Equation  is  one  in  which  some  or  all  of  the 
known  numbers  are  represented  by  letters. 

Example  1.     2x-\-a=:7a  —  x. 

The   problem   is   to   determine  a  number  x,  such  that  the 
equation  is  satisfied  for  all  values  of  a. 
Solution  :  1.  2  x  -{-  a  —  1  a  —  x. 

2.   Transposing,  Sx  =  6  a. 

8.   Dg:.  x  =  2a. 

Check  :  Does  2(2  a)  -\-a  =  1a  —  2a? 

Does  4a  -\-  a  =  7  a  — 2a?    Yes. 

This  solution  means  that  the  number  x  is  always  double  the  number  a. 

Example  2.     Determine  the  number  x  in  the  equation 

(p  _  cxf^  (a  -  cxy=  b(b  -  a). 
Solution  :  1.  (b  —  cx)^  —  (a  —  cxy  =  b(b  —  a). 

2.  Expanding,  (62  -2bcx-\-  c%2)_  (^2  _2acx  +  d^x^)  =b^-  ah. 

3.  .-.  62  _  2  hex  +  c2a;2  _  a2  +  2  acx  -  cH^  -  b^  -  ab. 

4.  Transposing  and  uniting  terms, 

5.  Factoring, 

6-    I>2c(a-6): 


2  acx  — 

2bcx  = 

=  ««- 

ah. 

2cx{a 

-6)  = 

-a(a  - 

-6). 

2cx{a 

-&)- 

a(a 

-h) 

2c{a- 

-6) 

2c(a 

-h) 

.*.  x  = 

a 

SIMPLE  EQUATIONS  201 

Check:  Does  (b  - —  Y  -  (a -^Y  =  b{b  -  a)  ? 

Does  62 _  a6  ^.  ^  _  rt-i  4.  a-i  _  ^  =  b^-ab  ?  Yes. 

4  4 

Note.  After  simplifying  the  eiiuation  until  it  takes  the  form  of  the 
•  ■<}uation  in  step  5,  divide  both  members  of  the  resulting  equation  by  the 
coefficient  of  x. 


EXERCISE  92 

Solve  the  following  equations  for  x : 

„,         s  ^^x^  —  bb  —  x2xb 

1.  3x-5a  =  2(a-hx).  12. = 

^         '  ax  a  a      X 

2.  5(x-3b)==3x-llb.  (n'mx\       f   ,     n«\ 

3.  a{3bx-2a)=b{2a-3bx).   ^^'   ^*'V2"^  VJ"'^!,'^'"^  4 j 

4.  5rx— 6  5  =  3(ric  — s). 

5.  ax  —  ac=^bx  —  be. 

6.  ax  —  V?  =  —b{b-\-x).  15. 

7.  T{x  —  r)-=  s (s -\-2  r  —  x). 


X 

— 

"  1 

2x 

3. 

X 

x—a 

3 

X- 

-4 

5m  — 

271 

3a;  +  4      5m  +  2n 

2a;-f  3a^3a;-f  46 

8.  c{x-(^  =  d(d^-x).  •  2x-3b      Sx-4.a' 

9.  x-l-^-=^  =  — .  17    a;4-n      a;-n^2(m+ny 

w.        m^  *  a;  — m     a;+??i       x^—ni^ 

,.     m2      1_1       7i2  2na;-3      -      9nx-{-2 

nx     m     n     mx  nx—1  3nx  —  \ 

a  —  x_10b  —  a       1  iq       ^  ^  a^  —  b^ 

XX.    — — ~~ —  —  — ~~ —  — —  •  At/. 


bbx         16  ax       3  a  x  —  a     x  —  b     b(x-~b) 

20    ^^ ~ ^  I  hx-\-a_2  I  <^^  —  ^ , 
6a;  ttic  abx 

2j    10a;-3a^9a;-5a»     3(a;-f-2a^ 
3  a  3  a"  a«         ' 

22    »  _  a  —  2  6ca;  _  5_x  _  S  ac  —  Sbx  —  9  a 
2         46c      ~6c  i26c 


202  ALGEBRA 


23. 
24. 
25. 


x  —  2  a     x-\-2  a     ar'  —  4a^ 
3x(a  —  b)      a  —  2b_a  —  b 
a^  —  6^  x  +  b       x—b 

2  X  3a  2ar^4-aa^ 


3a;  —  a      a;  +  2a      3ar^  +  5aa;  —  2a^ 


146.  Applied  Algebra.  Some  of  the  equation  problems  given 
up  to  this  point  illustrate  one  use  of  algebra;  an  unknown 
number  may  sometimes  be  found  by  means  of  an  equation  if 
numerical  relations  involving  the  number  are  known.  In 
order  to  solve  the  equations  that  arise,  skill  in  the  funda- 
mental operations  (§  16),  factoring,  and  fractions,  is  necessary. 

In  §  17  and  §  94,  another  application  of  algebra  is  illus- 
trated ;  a  formula  is  often  used  to  express  in  convenient  form 
a  rule  of  computation.  In  deriving  formulce  and  in  using 
them,  all  of  the  algebra  so  far  studied,  and  much  more,  fre- 
quently, is  necessary. 

147.  Deriving  and  Using  Formulae.  Three  examples  of  the 
methods  of  deriving  formulae  will  be  given,  besides  those  in 
§17. 

Example  1.  Derive  a  formula  for  the  total  area  of  the 
walls  of  a  room  in  terms  of   the  height,  length,  and  width. 

Solution  :  1.   Let  h  =  the  height  of  the  room  in  feet. 
Let  w  =  the  width  of  the  room  in  feet. 

Let  I  =  the  length  of  the  room  in  feet. 

Let  S  —  the  total  area  of  the  walls  in  square  feet. 

2.  Then,  hw  =  the  area  of  one  end  wall  in  square  feet. 

hw  =z  the  area  of  the  other  end  wall. 
hi  —  the  area  of  one  side  wall. 
hi  —  the  area  of  the  other  side  wall. 

3.  .'.  S^hvo^-hw^-hl^rhl  ^Ihw^'l  hi. 
.'.  8=2h{w  +  l). 

Thus,  if  the  height  is  9  feet,  the  width  14  feet,  and  the  length  18  feet, 
the  total  area  of  the  walls  is 

/S  =  2  X  9  X  (14  +  18)  =  18(32)  =  676  square  feet. 


SIMPLE  EQUATIONS  203 

Example  2.     Derive  formulie  for  two  numbers  whose  sum 
is  a  and  such  that  the  larger  exceeds  the  smaller  by  h. 
Solution  :  Let  s  =  the  smaller  number. 

Then,  s  -\-  b  =  the  larger  number. 

2.  .-.  s  +  (s  +  6)  =  a,  or  2  s  +  6  =  a. 

.-.  2s  =  a-6,  ors  =  ^^:i^. 

2 

The  smaller  number  is  ^  ~     and  the  larger  number  is  ^^-^t — 

25  —  7       18 

Thus,  if  a  is  25  and  6  is  7,  the  smaller  number  is or  — ,  or  9, 

2  2' 

and  the  larger  is  — ^  ,  or  16. 
id  , 

9  +  16  =  25  and  16  =  9  +  7. 

Example  3.  Derive  a  formula  for  the  rote  in  terms  of  the 
<  I  mount,  the  principal,  and  the  time. 

Solution  :  1.     The  fomiula  for  the  amount  (§  44)  is 

A  =  P  +  ^. 
100 

2.  Mioo  :  100  ^  =  100  P  +  Prt. 

3.  Sioop:  100  ^  -  100  P  =  Prt  =  {Pt)r. 

4  100^-lOOP^      ^^,.^100U-P). 

Pt  '  I't 

This  formula  enables  us  to  find  r  when  A,,  P,  and  t  are  known.    Thus, 
if  a  man  receives  §3500  at  the  end  of  6  years  from  an  investment  of 
$  2400,  what  rate  of  simple  interest  has  his  money  earned  ? 
Here,  A  =  %  3600,  P=$  2400,  and  «  =  6. 

.  ^  ^1^(3500  -  2400)  ^  1100  ^  7  g^  . 
-8406-  X  6  144         '       ' 

24 
that  is,  the  rate  is  7 .6  %. 

In  the  following  list  of  examples,  a  number  of  formulae, 
taken  from  physics,  chemistry,  geometry,  and  engineering,  are 
given.  It  is  impossible  to  show  in  this  text  how  these 
formulae  are  derived,  as  that  calls  for  special  knowledge  of 
these  various  subjects. 


204  *  ALGEBRA 

EXERCISE  93 

(a)  Express  each  of  the  first  6  formulae  in  words. 
(6)  Solve  each  of  them  for  the  letters  indicated. 

1.  ^  =  ab.     (a)  Solve  for  a;  (b)  for  b.  (§  17) 

2.  A=^'     (a)  Solve  for  a;  (b)  for  6.  (§17) 

3.  (7  =  27rr.     Solve  forn  (§17) 

4.  F=  Iwh.     (a)  Solve  for  w  ;  (b)  for  h.  (§  17) 

5.  F=  i  bh.     (a)   Solve  for  6;  (6)  for  7i.  (§  17 j 

6.  i^  =  |(7+32.  Solve  for  0.  (§84) 

7.  ^  =    ^         ^  is  a  formula  from  geometry. 

(a)  Find  ^  when  a  =  15 ;  ?>  =  24 ;  and  c  =  20. 
(6)  Find  c  when  ^  =  550 ;  6  =  30 ;  and  a  =  22. 
(c)  Solve  the  formula  for  a ;  (c?)  for  6. 

8.  A  =  F+?n. 

100 

(a)  Solve  for  P.     (5)  for  ^. 

9.  r=  -  +  ^  is  a  formula  from  physics. 

a 

(a)  Solve  it  for  t.     (b)  for  a. 

10.  mg  —  T=mfis  3.  formula  from  physics, 
(a)  Solve  it  for  T;  (b)  for/;  (c)  for  m. 

11.  u  =  ( — — — ]v  is  a  formula  from  physics. 

V3/H-  ml 


(a)  Solve  it  for  m;  (&)  for  iJf. 

O  = —  is  a  formula  from 

b  —  a 

(a)  Solve  it  for  a;  (b)  for  b. 


12.    O  =    ^^     is  a  formula  from  physics. 
b  —  a 


SIMPLE   EQUATIONS  205 

13.  s  =  ^^ —  is  a  formula  from  physics. 

b  —  a 

(a)  Solve  it  for  6;  (b)  for  a. 

14.  h  =  k(l  4-  ^Ij  t)  is  a  formula  from  physics, 
(a)  Solve  it  for  i;   (b)  for  k. 

T   JIf 

15.  I  = is  a  formula  from  physics. 

Mt 

(a)  Solve  it  for  L;  (b)  for  M. 

16.  w  = j5 —  is  a  formula  from  engineering. 

1  -f  — — 
600  fZ^ 

(a)  Find  u^  correct  to  two  decimals,  when  a  =  18,  1=12, 

and  cZ  =  10. 

(6)  Solve  the  formula  for  a. 

17.  p  =  —  +  d  is  a  formula  from  engineering. 

V 

(a)  Find  p  when  a  =  .56,  cZ  =  ^,  ^  =  f . 

(b)  Solve  the  formula  for  t. 

E 

18.  O  = is  a  formula  from  physics. 

R  +  - 

n 

(a)  Simplify  it  in  the  right  member. 

(b)  Solve  it  for  n ;  (c)  for  r. 

2 

19.  F= is  a  formula  from  physics. 

(a)  Find  F  when  m  =  150,  v  =  25,  f/  =  32  and  r  =  5. 
(6)  Solve  it  for  r ;  (c)  for  m. 

20.  -  =  -  +  -  is  a  formula  from  physics. 
f     P      <I 

(a)  Find  p  when  /=  30  and  q  =  40. 
(6)  Solve  it  forp;  (c)  for  q- 


XII.   GRAPHICAL   REPRESENTATION 

148.  Certain  mathematical  relations  may  be  represented  by 
means  of  geometrical  figures.  Paper  ruled  as  in  the  figures 
below  is  used ;  such  paper  is  called  Coordinate  Paper. 

Figure  1  is  an  illustration  of  this  manner  of  comparing  num- 
bers. The  lines  represent  the  lengths  of  some  of  the  rivers  of 
America.  The  side  of  one  square  represents  200  miles;. from 
this  fact,  the  lengths  of  the  rivers  may  be  determined. 

Such  representations  of  number  relations  are  called  Graphical 
Representations,  or  simply  Graphs. 

EXERCISE  94 

1.    Determine  the  lengths  of  the  rivers  in  Fig.  1. 


" 

1 

■2 

00 

nr,  ,   - 

■  i 

1 

Y(- 

-Mississipf 

e 

A 

(e 

d- 

- 

- 

■ 

' 

"" 

S 

t  La^wt^ 

rJ.  LL 

■|    1 

^ 

'"" 

~ 

j   1    1 

A  J-Lv 

,0 

e 

- 

- 

"1 

Oas.vl 

Arkansa 

1    i    1 

A.L  L.. 

1 

OasTgrpd 

-L,  rL 

-W-U^Jwio'- 

1 

^-Ire^tJwjto 

' 

u'     '- 

iJ 

e 

1  LLl.. 

Ik, 

_ 

1 

_ 

_j 

Fig.  1 


Fig.  2 


2.  In  Fig.  2,  the  weight  per  quart  of  certain  grains  is  repre- 
sented.    Find  the  weight  of  each  kind  of  grain. 

3.  Find  the  length  of  each  line  in  Fig.  3,  and  arrange  your 
results  in  the  form  of  a  table ;  when : 

206 


GRAPHICAL   REPRESENTATION 


207 


_ 

i 

it :.»._^        c_;d__e -f 

T              An--- e 

■"rh"  """  T"  "T  "T  ■"■        rU r"         " 

'  '  '_!       L 

\IL L  _ 

T?T       1 " 

^Y — 

'?^'                                              "^ 

1 

^^(^~ 

1      t                         1 

XY^H                                       i20|     _^           ^^       1       .                                ■    - 

1                                                  1 

Fig.  3 


Represent  by  means  of  lines  the  following  sets  of  statistics : 
4.    A  family  used  the  following  amounts  of  gas  during  the 
months  of  the  year: 


January  . 
February 
March 
April  .  . 
May  .  . 
June    .     . 


3600  cu.  ft. 
3200  cu.  ft. 
2000  cu.  ft. 
3100  cu.  ft. 
2900  cu.  ft. 
2900  cu.  ft. 


July  .  . 
August  . 
September 
October  . 
November 
December 


2300  cu.  ft. 
2200  cu.  ft. 
3100  cu.  ft. 
3200  cu.  ft. 
2800  cu.  ft. 
2200  cu.  ft. 


Hint.     Let  the  side  of  one  square  represent  100  cu.  ft.,  or  200  cu.  ft.; 
then  represent  the  various  given  numbers  by  lines  of  appropriate  length. 

5.   The  same  family  used  the  following  amounts  of  electricity 
during  the  months  of  the  year. 

January 27  k.w. 

February 27  k.w. 

March       24  k.w. 

April 19  k.w. 

May 15  k.w. 

June 9  k.w. 

A  k.w.  is  a  "kilowatt,"  the  unit  used  for  measuring  the  quantity  of 
electricity  consumed. 


July 

.     .     10  k.w. 

August      .... 

.     .       8  k.w. 

September      .    .     . 

.     .     18  k.w. 

October     .... 

.     .     42  k.w. 

November      .     .     . 

.     .     49  k.w. 

December      .     .     . 

.     .     45  k.w. 

208 


ALGEBRA 


6.   The  following  statistics  give  in  cents  the  cost  per  pound 
of  certain  common  articles  of  food  in  1910. 

Wheat 2.5 

Ry^ 3 

Navy  Beans 6 

Potatoes 1.3 

Rice 10 

Beafsteak 17 


Egg       .... 

...     20 

Bacon 

Milk 

.     .     .     22.5 
...       4 

Apples  .... 
Oatmeal     .... 

...       5 
...     10 

Buckwheat     .     .     . 

...      2.5 

7.   The  average  hourly  velocity  of  wind  in  miles  at  the  fol- 
lowing points  in  the  United  States  is : 


Boise,  Idaho 4 

Atlanta,  Ga 9 

Buffalo,   N.Y 11 

Duluth,  Minn 7 

Omaha,  Neb.     ......  8 


Portland,  Me .  6 

Vicksburg,  Miss 6 

Philadelphia,  Pa 10 

Chicago,  111 9 

Boston,  Mass 11 


8.   The  average  annual  precipitation  (rain  and  melted  snow), 
in  inches,  at  the  following  cities  is : 


San  Diego,  Cal 10 

Denver,  Colo 14 

Springfield,  111 37 

Dubuque,  la 34 

Baltimore,  Md 43 


Mobile,  Ala 62 

Flagstaff,  Ariz 23 

Salt  Lake  City,  Utah   ...  16 

Wilmington,  N.C 51 

Portland,  Me. 42.5 


149.  In  drawing  the  graph  of  a  set  of  statistics,  the  paper 
is  usually  prepared  as  in  Fig.  4.  Two  lines  are  drawn  at  right 
angles.  The  numbers  to  be  represented  are  examined  in  order 
to  decide  what  number  shall  be  represented  by  the  side  of  one 
square.  The  number  represented  by  one  side  of  a  square 
is  called  the  Unit. 

Instead  of  drawing  full  lines  to  represent  the  various  num- 
bers, it  is  customary  to  place  points  where  the  lines,  of  proper 
length  would  end.  Finally  these  points  are  connected  by  a 
smooth  line.  One  line,  that  at  11  a.m.,  is  drawn  in  full  in  this 
graph. 


GRAPHICAL   REPRESENTATION 


209 


Example  1.     Draw  a  graph  representing  the  temperature 
readings  at  the  hours  indicated. 

11  A.M.     .  .  4-    7°  4  P.M.  .    .    .  +  9° 

12  M.     .  .  +    8°  5  P.M.  .     .     .  +8° 

1  P.M.     .  .  -I-    9°  6  P.M.  ...  +6° 

2  P.M.     .  .  +  10°  7  P.M.  .     .     .  +4° 

3  P.M.     .  .  + 10"  8  P.M.  ....  0 


Oa.m.  ...  -8° 

7  A.M.  ...  -6° 

8  A.M.  ...  -2° 

9  a.m.  ...  +2° 
10  a.m.  ...  +6° 


L 

-                                                               -                     s 

:^=  ::„:::: :,::«i-,: :_: ::,_::. .^^l 

::  ^5 ::::::  —  ::z_:: :: iz      z    ztsz  z 

z                                .    __  _t  s 

:    :                  ~z         ~ :_    :    t  z\ 

:                       z           _                                  ji  -  ui. 

IE .i ^   '   _-i.lJii.S-- 

_    ^  _  J  ^ 8/    -?       0     1        I      1       1-     ;    -  -.'               >     -1 

^_                        .  _                      __j_ 

__<!''                        _                       ___                        -_                                    on^c  egred 

"0           III                      1                             'i''"»  1  r  T  ^°" 

Fig.  4 


Notice  that  the  units  of  time  and  of  temperature  are  indicated  ;  that 
the  line  upon  which  the  hours  are  indicated  is  placed  near  the  center  of 
the  paper  to  allow  for  the  negative  temperatures  ;  that  the  vertical  line  is 
placed  at  the  left  edge  to  allow  space  for  all  of  the  hours  of  the  day  ;  that 
the  line  connecting  the  points  is  made  a  smooth  curve.     Review  §  24. 


150.   The  graph  in  Fig.  4  may  be  used  to  illustrate  another 
advantage  of  graphical  representation. 

A  graph  may  assist  in  finding  new  information. 

What  was  the  approximate  temperature  at  6.30  p.m.  ? 
Solution  :  1.   On  the  graph,  the  point  8  would  indicate  the  time  6.30. 

2.  The  point  i?,  above  it  on  the  graph,  indicates  the  temperature. 

3.  II  is  opposite  +  5°  on  the  vertical  line. 

4.  The  approximate  temperature  then  at  6.30  p.m.  was  6°. 


210  ALGEBRA 


EXERCISE  95 

1.  Draw  a  graph  representing  the  following  temperature 
readings : 

6  A.M +12^            1  P.M +16° 

7  A.M +14°          2  P.M +12° 

8  A.M +18°          3  P.M +6° 

9  A.M +20°          4  P.M 0° 

10  A.M +22°  5  P.M. -    4° 

11  A.M +22°  6  P.M -    6° 

12  M +19°  7  P.M -    7° 

2.  From  the  graph  in  1,  find  the  approximate  temperature 
at  7.30,  8.30,  11.30,  2.30,  5.30. 

3.  Get  the  temperature  readings  in  your  own  school  district 
to-morrow,  and  draw  the  graph. 

4.  In  order  to  protect  his  orchard  from  a  frost,  a  man  tried 
to  warm  the  air  in  the  orchard  by  burning  oil  heaters.  Below 
are  given  the  temperature  readings  within  the  heated  area  and 
outside  of  it.  Plot  the  two  sets  of  readings  on  the  same 
sheet,  making  one  line  for  the  readings  within  the  heated  area, 
and  one  for  the  other  readings. 


Tun 

Tempeeatcbe 

TiMB 

Tempeeatuke 

Inside 

Outside 

Inside 

Outside 

8.00 

35° 

35° 

10.30 

36° 

32° 

8.15 

38° 

35° 

11.00 

35° 

31° 

8.30 

39° 

33° 

11.30 

36.5° 

30° 

8.45 

40° 

32° 

12.00 

36.5° 

28° 

9.00 

40° 

34° 

12.30 

34.5° 

27° 

9.30 

40° 

34° 

1.00 

34.6° 

27° 

LO.OO 

40° 

34° 

5.  Savings  banks  and  trust  companies  pay  compound  inter- 
est on  money  left  with  them ;  that  is,  interest  upon  interest. 

The  following  table  shows  the  amount  of  money  which 
must  be  saved  annually  at  4  %  compound  interest  to  amount 


GRAPHICAL   REPRESENTATION  211 

to  SIOOO  in  a  definite  number  of  years.     (To  simplify   the 
problem  even  dollars  are  taken.) 

10  years .$80      25  years 8  23 

15  years .$48      30  years «17 

20  years $32      35  years $13 

6.  Assume  that  a  train  leaves  station  A  at  1  p.m.  and  that 
it  arrives  at  F  and  J,  the  distances  of  which  from  A  are  given, 
at  the  times  indicated.  Draw  the  graph  representing  the  time 
when  the  train  reaches  points  between  these  three  stations, 
making  the  graph  a  broken  line,  (/  )  From  the  graph,  de- 
termine when  the  train  reaches  the  other  points,  thus  making 
a  time-table  for  the  train  between  A  and  H. 

A.  0  miles.     1p.m.  F.     60  miles.     3  p.m. 

B.  6  miles.  G.     80  miles. 

C.  15  miles.  H.    104  miles. 
D-   24  miles.  I.    120  miles. 

E.   42  miles.  J.    140  miles.     5  p.m. 

Hint.  If  possible,  lay  off  the  distances  on  the  horizontal  line,  making 
the  unit  2  or  6  miles,  and  the  time  on  the  vertical  line,  making  the  unit 
6  or  12  minutes. ) 

7.  Complete  the  following  table  showing  the  number  of  dol- 
lars simple  interest  at  6  %  on*  $  100  for  the  number  of  years 
indic9,ted  and  draw  the  graph. 

1  year  ;  interest  $6.  5  years;  interest  ? 

2  years ;  interest  $  12.  6  years ;  interest  ? 

3  years  ;  interest  ?  7  years ;  interest  ? 

4  years  ;  interest  ?  8  years ;  interest  ? 

8.  From  the  graph,  determine  the  interest  for  5  years  and 
6  months. 

151.  Certain  terms  used  in  mathematics  in  connection  with 
graphical  representation  will  now  be  given.  Referring  to  Fig. 
6  below :  the  lines  XX'  and  YY',  drawn  at  right  angles,  are 
called  Axes :  XX',  the  Horizontal  Axis,  and  YY',  the  Vertical 
Axis  ;  the  point  0  is  called  the  Origin.    From  P,  perpendiculars 


212 


ALGEBRA 


are  drawn  to  the  axes ;  the  line  PR  is  called  the  Ordinate  of  P, 
and  the  line  PS  is  called  the  Abscissa  of  P ;  together  they  are 
called  the  Coordinates  of  P.  The  axes  are  called  Coordinate 
Axes.     On  the  axes,  the  scale  used  is  indicated.     The  distances 


~ 

~ 

- 

~ 

~ 

~ 

r 

- 

~ 

- 

~ 

- 

~ 

~" 

~~ 

' 

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r 

i'^ 

r, 

' 

-  ll 

n 

" 

-Is 

^ 

J, 

., 

;, 

'- 

^ 

X 

L 

U 

X 

-( 

- 

-i 

\ 

. 

? 

- 

) 

- 

♦ 

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j) 

t^ 

+< 

♦  s 

♦ 

s 

♦ 

1 

' 

i* 

J* 

_ 

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M 

Fig.  5 


on  OX  are  positive,  on  OX'  are  negative;  on  OF  are  positive, 
and  on  OY'  are  negative.  The  abscissa  of  P  is  3  and  the 
ordinate  is  4 :  the  point  P  is  called  The  Point  (3,  4).  Notice 
that  the  abscissa  is  written  first,  and  the  ordinate,  second. 


EXERCISE  96 

1.  What  is  the  abscissa  of  the  point  ^  ?   5?    O?   D7 

2.  What  is  the  ordinate  of  the  point  ^?   B?    G?   D? 

3.  What  are  the  coordinates  of  jE?  F?    G?   H? 

4.  Prepare  a  piece  of  graph  paper  similar  to  that  used  for 
Fig.  5.  Locate  the  points :  (3,  6) ;  (4,  2) ;  (5,  -  3) ;  (-  2,  5) ; 
(~4,  ^6). 


GRAPHICAL  REPRESENTATION 


213 


5.  Locate  the  points  A:(l,  1);  B :  (2,  6)  ;  C:  (11,  6); 
Z>:(10,1). 

Draw  the  lines  AB,  BC,  CD,  and  AD.  What  figure  is 
formed  ?  Draw  the  diagonals  of  the  figure,  and  find  the  co- 
ordinates of  the  point  where  they  meet. 

6.  Locate  the  points  ^  :  (3,  -  5)  ;  J5  :  (4,  -  2)  ;  C:  (7,  -  2) 
D :  (S,  -  5) ',  E  :  (7,  -  S)  :  F:  (4,  -  8).  Draw  the  lines  join- 
ing the  points  in  order.     What  figure  is  formed  ? 

152.  The  numbers  which  are  represented  graphically  in 
mathematics  are  usually  connected  by  an  equation. 

Example.     Draw  the  graph  of  the  equation 

2^4 
Solution  :  1.    Solve  the  equation  for  y. 
M4  :  2x  +  y  =  i. 

S2x:  y=4-2x. 

2.  Select  any  value  of  x  and  find  the  corresponding  value  of  y.  Thus, 
if  X  =  1,  y  =  4  —  2  =  2.     Similarly  : 


when 

X  = 

-  5 

-   8 

-1 

0 

+  2 

+  4 

+  6 

+    7 

then 

y  = 

+  14 

+  10 

+  0 

+  4 

0 

-  4 

-8 

-10 

.3.  Use  the  pairs  of  numbers  so  obtained  as  coordinates  of  points  ;  thus, 
locate  the  points  (-5,  +  14)  ;  (—3,  +  10)  ;  (-1,  +  6)  ;  etc. 
4.    Draw  the  line  connecting  the  points.     (Graph  on  page  214.) 
Notice  that  the  graph  seems  to  be  a  straight  line.     The  coordinates  of 
the  point  A  on  the  graph  are  +3.5  and  —  3.     Do  these  satisfy  the  equa- 
tion ? 

Does  M  +  :i3  =  i? 
2         4 

Does  1.75+(- .75)  =  1?    Yes. 


214 


ALGEBRA 


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n 

^ 

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.,. 

s 

,..,. 

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r 

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■ 

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s 

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V' 

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1 

Fig.  6 


EXERCISE  97 

1.  Determine  the  coordinates  of  B,  C,  and  D.  Determine 
whether  they  satisfy  the  equation  of  the  graph  ? 

2.  Select  any  point  not  on  the  graph,  find  its  coordinates, 
and  determine  whether  they  satisfy  the  equation  of  the  graph. 

3.  Similarly,  draw  the  graph  of  y  =  x-\-S.  For  x  select  the 
values  -1,  -  2,  -  3,  -  4,  +  1,  +  2;  -f  3,  +  4,  0.  What  does 
the  graph  appear  to  be  ? 

4.  Select  three  new  points  which  are  on  the  resulting  graph, 
find  their  coordinates,  and  determine  whether  they  satisfy  the 
equation  given  in  Example  3. 


5.   Draw  the  graph  of  ^ 


1. 


Select  at  least  four  negative  and  four  positive  values  of  x, 
and  from  them  determine  the  corresponding  values  of  y. 
What  sort  of  graph  do  you  obtain  ? 


GRAPHICAL   REPRESENTATION  215 

6.  Select  any  three  points  on  the  graph  and  determine 
whether  their  coordinates  satisfy  the  equation. 

153.  The  equations  in  the  preceding  paragraph  have  each  had 
tn:o  unknown  numbers.  The  following  facts  are  to  be  remem- 
bered from  the  examples  of  §  152 : 

1.  The  graph  of  an  equation  of  the  first  degree  (§  77)  having  two 
unknowns  is  a  straight  line.  For  this  reason,  first  degree  equations 
are  also  called  linear  equations. 

2.  The  coordinates  of  every  point  on  the  line  satisfy  the  equation. 

3.  The  coordinates  of  every  point  not  on  the  line  do  not  satisfy 
the  equation. 

154.  A  Solution  of  an  equation  having  two  or  more  unknowns 

is  a  set  of  values  of  the  unknowns  which  satisfy  the  equation. 

Be  careful  not  to  confuse  the  word  "solution  "  in  the  sense  of  this  para- 
graph with  the  same  word  when  it  is  used  to  mark  the  process  of  solving  an 
example  as  is  done  in  the  text. 

Example.     Consider  the  equation  x  +  y  =  5. 

(1,  4)  is  a  solution  because  1+4  =  5. 

(—  8,  +  18)  is  a  solution  because  —  8  +  13  =  5. 

155.  Number  of  Solutions.  There  are  an  indefinitely  large 
number  of  points  upon  a  straight  line.  This  is  expressed  by 
saying  that  there  are  an  injinite  number  of  points  on  a  straight 
line.  Since  the  coordinates  of  each  point  satisfy  the  equation 
of  the  line,  then :  there  are  an  infinite  number  of  solutions  of  a 
linear  equation  ivith  two  unknowns. 

This  fact  is  evident  also  because  for  every  value  of  one  of 
the  unknowns,  a  value  of  the  other  may  be  found. 

Example.     Consider  the  equation  2x-\-3y=zl5, 
When  x  =  1,  2  .  1  +  3  2/  =  16  or  3  y  =  13,  2/  =  4| ; 
then  a-  =  1,  y  =  4|  is  one  solution. 

Similarly  when, 

x=2;     y  =  i^.  x=-S;  y  =  +  7. 

x  =  3;    y  =  3.  x=-5,  y=  +  S\. 


216  ALGEBRA 

Thus  as  X  changes  in  value,  in  such  an  equation,  y  also 
changes  in  value,  acquiring  a  new  value,  x  and  y  are  said  to 
vary,  and  are  called  Variables.  Hereafter  these  equations  will 
be  called  equations  having  two  variables. 

An  equation  of  the  first  degree  having  two  variables  has  an  in- 
finite number  of  solutions. 

Such  equations  are  called  Indeterminate  Equations. 

166.  The  graph  of  an  indeterminate  linear  equation  with  two 
variables  is  always  a  straight  line  by  §  153.  A  straight  line 
may  be  drawn  with  a  ruler  as  soon  as  two  of  its  points  are 
known.     This  leads  to  the 

Rule.  —  To  draw  the  graph  of  a  linear  equation  having  two  var- 
iables : 

1.  Select  one  value  for  one  variable  and  determine  the  correspon- 
ding value  of  the  other  variable ;  this  gives  one  solution. 

2.  Determine  a  second  solution  as  in  step  1. 

3.  Plot  the  two  points  whose  coordinates  are  the  pairs  of  numbers 
and  connect  them  with  a  straight  line. 

4.  Check  the  result  by  finding  a  third  solution,  and  plotting  the 
corresponding  point.  The  third  point  should  fall  upon  the  graph  ob- 
tained in  step  3. 

Example.     Draw  the  graph  oiAx  —  Sy  =  6. 

Solution  :  1.    Let  x  =  0  ;  then  y  =—2. 

(4.0~3y  =  6;    -  3  ?/ =  6  ;   i/=-2.) 

2.  Let  x  =  S',   then  y  =  2. 

(4.8-3!/  =  6;    -3?/  =  6-12;    -Sy=-6;   y  =  2.) 

3.  Plot  the  points  and  draw  the  line.     See  Fig.  7. 

Check  :  Let  x  =  Q  ;  then  y  =  6.     Is  this  point  on  the  line  ? 

Note.  To  get  the  best  results,  it  is  necessary  to  use  coordinate  paper  for 
the  following  exercises.  Fair  results  may  be  obtained  by  using  paper  ruled 
by  the  pupil. 


GRAPHICAL  REPRESENTATION 


217 


Fig.  7 


EXERCISE  98 

Draw  the  graphs  of  the  following  equations,  each  on  a^  sepa- 
rate sheet  of  graph  paper : 


1.  a;  +  2/  =  5. 

2.  2x  —  y  =  6. 

3.  2x-\-Sy  =  6. 

4.  3x-2y  =  12, 

5.  5  a? -f  4  2/ =  20. 


6.  Sx  —  5y=15. 

7.  7  a;  4- 4?/ =  2. 

8.  5x  —  Sy  =  —6. 

9.  x-6y=-10. 
10.  Sx=5-7y. 


157.   Independent  Equations  and  Simultaneous  Equations. 

Example.     Draw  upon  the  same  sheet  the  graph  of 

2x-y==4..  (1) 

2x-\-Sy  =  12. 
Solution  :  1.    For  equation  1 :  2  x  —  y  =  4. 
Ifx  =  0,  y=-4;ify  =  0,  5c  =  -f-2. 
Solutions :    (0,  -  4)  and  (+  2,  0). 


(2) 


218 


ALGEBRA 


2.  For  equation  2  :  2x  +  Zy  =  l2. 
li  X  =  0,  y  =  4  ;   if  y  =  0,  x  =  6. 

Solutions  :  (0,  +  4)  and  (  +  6,  0). 

3.  See  Fig.  8  for  the  graphs. 


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3.8 

Questions  :  1.  What  is  true  about  the  coordinates  of  all  points  on 
line  1  ?     (§  153.) 

2.  What  is  true  about  the  coordinates  of  all  points  on  line  2  ? 

3.  What  then  is  true  of  the  coordinates  of  point  A  ? 

4.  Are  there  any  other  points  whose  coordinates  will  satisfy  both 
equations  ? 

Two  equations,  each  having  two  variables,  are  called  In- 
dependent Equations  when  each  has  solutions  which  do  not 
satisfy  the  other.  Thus,  the  equations  above  are  independent. 
Their  graphs  are  two  different  straight  lines. 

Two  linear  equations  which  are  independent  and  which  have 
one  common  solution  are  called  Simultaneous  Linear  Equations. 
Their  graphs  cross  at  one  point.  The  above  equations  are 
simultaneous  since  they  have  the  common  solution  (3,  2). 


GRAPHICAL  REPRESENTATION  219 

Rule.  —  To  determine  graphically  the  common  solution  of  two 
simultaneous  linear  equations  having  two  variables  : 

1.  Draw  upon  one  sheet  the  graphs  of  both  equations. 

2.  Find  the  coordinates  of  the  point  common  to  the  two  lines. 
This  is  the  common  solution. 

3.  Test  the  common  solution  by  substituting  it  in  both  equations. 

Historical  Note.  One  of  the  notable  advances  in  mathematics  is 
intimately  associated  with  the  subject  of  this  chapter.  The  idea  of  rep- 
resenting some  geometrical  figures  by  algebraic  expressions  occurred  to 
mathematicians  early.  In  the  14th  century,  Oresme,  a  French  mathema- 
tician, discussed  certain  mathematical  ideas  by  means  of  coordinates. 
He,  however,  used  only  positive  coordinates,  thus  confining  himself  to 
what  is  called  the  first  quadrant. 

To  the  French  mathematician,  Descartes  (1596-1650),  is  due  the  ex- 
tension of  the  use  of  coordinates.  He  considered  negative  as  well  as 
positive  coordinates  and  also  discussed  two  or  more  lines,  or  curves, 
drawn  with  respect  to  the  same  system  of  coordinate  axes.  His  improve- 
ment brought  with  it  a  better  understanding  of  negative  numbers  and  of 
negative  roots  of  equations,  and  laid  the  foundation  for  Analytic  Geom- 
etry, one  of  the  subjects  of  great  interest  to  mathematicians. 

EXERCISE  99 

Determine  graphically  which  of  the  following  pairs  of  equa- 
tions are  simultaneous,  and  which  are  dependent;  if  simulta- 
neous, find  their  common  solution  : 


6.  r  2  a;  -f  2/  =  4. 

9.  *    1  6a; -f  3  2/ =  12. 

10. 


1.    |«  +  ^ 
\a—  0 

\  Z  X  —  y  =:  b.  \x  -—  y  =  *d. 

(x  +  2y:^r. 
•*•    \Sx-y  =  l. 


{3a -2b  =  6. 
6  a  -  4  6  =  12. 


(3x-2y^-12.  i  5x-6y  =  S. 

{Ax-\-2y  =  -2.  '    [9x-4:y=z-6. 


220  ALGEBRA 

9.   Draw  on  one  sheet  the  graphs  of  the  following  equations 
and  thus  determine  whether  they  have  a  common  solution : 
(a)  5  a-  -  2  2/  =  7 ;   {b)  3  a;  +  2  y  =  17  ;  (c)  a;  -  3  y  =  -  9. 

lb.  Determine  whether  the  following  equations  have  a 
common  solution : 

(a)  2x-y  =  S',  (b)  2x-  y=A',  (c)  2x-\-3y  =  20. 

158.  Inconsistent  Equations.  Two  equations  may  be  inde- 
pendent (§  157),  and  yet  not  be  simultaneous;  that  is,  they 
may  not  have  a  common  solution.  The  graphs  of  two  simul- 
taneous linear  equations  are  two  intersecting  straight  lines. 

Two  independent  linear  equations  which  do  not  have  a  com- 
mon solution  are  called  Inconsistent  Equations.  Their  graphs 
are  parallel  lines. 

EXERCISE   100 

Determine  graphically  which  sets  of  equations  are  inconsis- 
tent; if  simultaneous,  determine  the  common  solution. 


x  —  3y  =  6. 
2x-6y  =  -lS. 


4m-3n  =  12. 
16  771  - 12  71  =  24. 


[5  a- 26  =  10.  \2x-5y  =  -16. 

2.       .^  ^,  ^.  5.     ' 


3. 


15a-66  =  -24.  [3x-\-7y  =  5. 

I2x 
Ux 


2x-Sy  =  lS.  l2x-{-Sy  =  6, 

x-^3y  =  0.  '    l4ic  +  62/  =  8. 


159.  Sometimes  it  is  difficult  to  find  the  common  solution 
accurately  by  this  graphical  method.  As  an  example,  find  the 
common  solution  of  the  pair  of  equations : 

(1)  5x-9y  =  ll',    (2)  3x-\-7y  =  9. 

In  the  next  chapter,  other  methods  of  solving  simuitaneous 
linear  equations  are  given. 


XIII.   SIMULTANEOUS  LINEAR  EQUATIONS 


DEFINITIONS 

160.  If  a  rational  and  integral  monomial  (§  113)  involves 
two  or  more  letters,  its  degree  with  respect  to  them  is  denoted 
by  the  sum  of  their  exponents. 

Thus,  2a:^hxy^  is  of  the  fcnirth  degree  with  respect  to  x  and  y. 

161.  If  each  term  of  an  equation  containing  one  or  more  un- 
known numbers  is  rational  and  integral,  the  Degree  of  the  Equa- 
tion is  the  degree  of  its  term  of  highest  degree. 

Thus,  if  X  and  y  represent  unknown  numbei-s, 

ax  —  &y  =  c  is  an  equation  of  the  first  degree  ; 
X*  4-  4  X  =  —  2  is  an  equation  of  the  second  degree  ; 
2  x2  —  3  xy^  =  5  is  an  equation  of  the  third  degree. 

162.  An  equation  of  the  first  degree  is  also  called  a  Linear 
Equation. 

163.  A  Solution  of  an  equation  with  two  unknowns  is  a  set 
of  values  of  the  unknowns  which  together  satisfy  the  equation. 

Thus  X  =  12  and  y  =  2  is  a  solution  of  x  +  2  y  =  16. 

A  linear  equation  wdth  two  unknowns  has  an  ii^finite  number 
of  solutions.  A  solution  is  obtained  by  assigning  any  value  to 
one  unknown  and  finding  the  corresponding  value  of  the  other 
unknown.     Thus : 

(a)  Some  of  the  solutions  of  the  equation  x  +  2  y  =  16  are  : 


x  =  +  l 
2/  =  +  7.5 

4-2 
+  7 

-4 

+  10 

+  10 

+  3 

+  20 
-2 

(6)  Some  of  the  solutions  of  the  equation  x  —  2  y  =  4  are : 


x  =  +l 
y=-1.5 

+  2 
-  1 

-4     +10 
-.4     +3 

+  20 
+  8 

221 


222  ALGEBRA 

An  equation  with  two  unknowns  is  called  an  Indeterminate 
Equation,  and  the  unknowns  are  called  Variables. 

164.  Two  linear  equations,  each  containing  two  variables, 
are  said  to  be  Independent  Equations  if  each  has  solutions  which 
are  not  solutions  of  the  other. 

The  two  equations  in  §  163  are  independent.    Why  ? 

The  equations  x  -\-y  =  5  and  2x  -\-2y  =  10  are  not  independent ;  for 
the  second  equation  can  be  reduced  to  the  form  of  the  first  by  dividing 
each  term  by  2  ;  hence  every  sohition  of  one  equation  is  also  a  solution 
of  the  other.     They  are  called  dependent  equations. 

165.  Two  independent  linear  equations  having  one  common 
solution  are  called  Simultaneous  Equations. 

The  two  equations  in  §  163  are  simultaneous.     Why  ? 

166.  Two  independent  linear  equations  which  do  not  have 
any  common  solution  are  called  Inconsistent  Equations. 

The  equations  a;  -}-  2  2/  =  16  and  2  x  +  4  ?/  =  21  are  inconsistent. 

167.  To  Solve  a  set  of  simultaneous  equations  is  to  find  their 
common  solution. 

Example  1.     Solve  the  equations    „         .         r.  /r»x 

Solution  :  \.   M^*  {\)  :  20  x  -  12  ?/  =  76.  (3) 

2.  Mi  (2)  :  21  a;  +  12  y  =  6.  (4) 

3.  Add  (3)  and  (4)  :  41  a;  =  82.  (5) 

4.  Z>4i  (5)  :  x  =  2.  (6) 

5.  Substitute  this  value  of  a;  in  (1)  :  10  -  3  y  =  19.  (7) 

-3y  =  9.  (8) 

y  =  -  3.  (9) 

6.  The  common  solution  is  x  =  2,  ?/  :=  —  3. 
Check  :  Substitute  in  (1)  :                     10  +  9  =  19. 
Substitute  in  (2)  :                                  14  -  12  =  2. 

*  See  §  42  for  the  symbol  J/4.  Read  this  "  multiply  both  members  of  equa- 
tion (1)  by  4." 


SIMULTANEOUS  LINEAR  EQUATIONS  223 

Notice  :  1.  That,  in  equations  (1)  and  (2),  the  coeflScients  of  y  are  not  the 
same. 

2.  That,  by  multiplications,  the  coefficients  of  y  are  made  the  same  in 
absolute  value  in  equations  (3)  and  (4). 

3.  That,  by  addition,  12  y  disappears  in  equation  (5),  and  that  20 x  and  21  a; 
are  combined,  giving  41  x.  This  is  allowable  because  we  assume  that  t  and  y 
represent  the  same  numbers  respectively  in  all  of  the  equations,  namely,  the 
particular  numbers  which  together  form  the  common  solution  of  equations 
(1)  and  (2).    y  is  said  to  be  Eliminated. 

Elimination  means  to  cause  to  disappear ;  thus  y  was  made  to  disappear 
by  adding  the  two  equations  (3)  and  (4). 

4.  That  the  remaining  number,  x,  is  then  easily  found. 

The  solution  is  an  example  of  Elimination  by  Addition. 


Example  2.     Solve  the  equations  j  1'^^  «  +  ^  6  _  1. 

(1) 
(2) 

Solution  :  1.    Mj  (1)  :                             30  a  +  16  ft  =  2. 

(3) 

2.    Ma  (2):                                                    30  a  -  21  6  =  -  72. 

(4) 

3.    Subtract  (4)  from  (3)  :                                   37  6  =  74. 

(5) 

4.    D37  (5)  :                                                                  6=2. 

(6) 

6.    Substitute  the  value  of  6  in  (1)  :           16  a  +  16  =  1. 

(7) 

.-.  15  a  =  -15. 

(8) 

.-.  a=-l. 

(9) 

The  solution  is:                                          a  =  —  l;?)  =  +2. 

Substitute  the  values  of  a  and  h  in  equations  (1)  and  (2). 

The  solution  is  an  example  of  Elimination  by  Subtraction. 

Rule.  —  To  solve  two  simultaneous  linear  equations  having  two 
variables  by  the  addition  or  subtraction  method  of  elimination : 

1 .  Multiply,  if  necessary,  both  the  first  and  second  equations  by 
such  numbers  as  will  make  the  coefficients  of  one  of  the  variables  of 
equal  absolute  value. 

2.  If  the  coefficients  have  the  same  sign,  subtract  one  equation 
from  the  other ;  if  they  have  opposite  signs,  add  the  equations. 

3.  Solve  the  equation  resulting  from  step  2  for  the  other  variable. 

4.  Substitute  the  value  of  the  variable  found  in  step  3  in  any  equa- 
tion containing  both  variables,  and  solve  for  the  remaining  variable. 

5.  Check  the  solution  by  substituting  it  in  both  of  the  original 
equations. 


224 


ALGEBRA 


Historical  Note.  Little  progress  was  made  in  solving  linear  equa- 
tions having  more  than  one  unknown  until  the  latter  part  of  the  15th  cen- 
tury, although  mathematicians  before  that  time  had  considered  such 
problems.  After  Stifel  and  Stevin  had  introduced  somewhat  simple  no- 
tations for  several  unknowns  and  their  powers,  definite  methods  for  sol- 
ving equations  of  the  first  degree  with  two  unknowns  were  developed. 
Johannes  Buteo,  a  French  monk,  (1492-1572),  solved  equations  with 
three  unknowns  in  a  text  on  algebra  which  appeared  in  1559. 


EXERCISE   101 
Solve  by  the  method  of  addition  or  subtraction : 


1. 


3. 


5. 


6. 


7. 


8. 


9. 


10. 


11.    i 


\Sx-\-y  =  ll. 
[5x-y  =  lS. 

[5a-hb  =  7. 

|r- 6  s= -10. 

\2r-7s=-15. 

f5  m-l- 4  71  =  22. 
13  m  +  n  =  9. 

[7c-2d!  =  31. 
4c-3d  =  27. 

5a  +  3b=-9. 
3a_46=-17. 

6x-h2y=-3. 
5x  —  3y=  —  6. 

|3s-|-7^  =  4. 
[7  s-\-St  =  26. 

[Sp-6q  =  10. 

{5r-9y  =  l, 
i8r-102/=-5. 

f8c-h9^  =  ^. 


12. 


13. 


14. 


15. 


16. 


17. 


18. 


19. 


20. 


l8c-9^  =  77. 


6x 
15 

~lly=-4 

x-^4.y  =  53. 

3x 

+  7y  =  2. 

.7x 

+  82/=-2. 

4.V 
14 

+  15(/  =  7. 

?;  4-6^  =  9. 

|28 

x-3Qy=:l. 

114 

a;  +  15  2/=6. 

i7x 

+  92/  =  8. 

19  a; 

-  8  2/  =  69. 

2x 

-f2/  =  13. 

X     y_U, 
[3     5     15 

\i' 

^2  =  .. 

s 
4" 

1-*- 

fl^^ 

.  +  ^  =  -1-. 

m 

.2 

^^_1    5 

-  7  -Itt. 

^ 

■I='- 

2a 
[  9 

«=!■ 

4. 


SIMULTANEOUS  LINEAR  EQUATIONS  225 

168.   Elimination    by    Substitution    is    a    second     common 
method  of  solving  simultaneous  equations. 

Example.     Solve  the  equations  I  „  ^  ~  ^  ^  ~~    ^'  ^J 

Solution  :  1.   Solve  (1)  for  x  in  terms  of  y  : 

7x  =  15  +  9i/;x  =  (l^i^V  (3) 

2.   Substitute  this  value  of  a;  iu  (2) : 


8y-5(^^  +  ^n  =  -I7. 


(4) 


3.  M7(4):  66y- 5(15 +  9y)  =  - 119.  (5) 

4.  Expanding:  5(3  ?/ -  75- 45y  =- 119.  (6) 

5.  Combining :  11  y  -  75  =-  119.  (7) 

6.  A75:  11 2/ =-44. 

7.  Dn:  2/ =-4. 

8.  Substituting  the  value  of  y  in  (3)  : 

a.^l5  +  9(-4)_15-36^ -21^     3 
7  77' 

9.  The  solution  is :  x  =  —  S,  y  =  —4.  Check  it  by  substitution  in  equa- 
tions (1)  and  (2). 

Note.  In  step  3,  when  multiplying  5f  ^'  '0  ^Y  '^>  ^^"^  obtains 
7  . 5(15 +  0y) .    ^jjg  ^.g  cancel,  giving  the  result  5(15  +  9 1/) . 

Rule.— To  solve  two  simultaneous  linear  equations  having  two 
variables  by  the  substitution  method  of  elimination : 

1.  Solve  one  equation  for  one  variable  in  terms  of  the  other  vari- 
able. 

2.  Substitute  for  this  variable  in  the  other  equation  the  value 
found  for  it  in  step  1. 

3.  Solve  the  equation  resulting  in  step  2  for  the  second  variable. 

4.  Substitute  the  value  of  the  second  variable,  obtained  in  step  3, 
in  any  equation  containing  both  variables  and  solve  for  the  first 
variable. 

6.  Check  the  solution  by  substituting  it  in  the  original  equations. 


226  ALGEBRA 

Historical  Note.     The  earliest  use  of  the  Substitution  Method  of 

Elimination  in  print,  of  which  we  have  any  record,  is  in  Newton's  Arith- 
metica  Universalis,  in  1707. 

EXERCISE  102 
Solve  by  the  substitution  method : 

l4?/i  +  w  =  16.  '  ll2p-hl0g  =  -5. 

2     |r-6s  =  2.  ^^  j^6w-9x  =  19. 

[3s-Sr==29.  '  ll5w+7a;  =  -41. 

g     \5x-\-Sy  =  +  2.  ^3  |15^-125  =  -54. 

[10  a.' -12  2/ =  +32.  '  110^+9jB  =  -2. 

|7a.--42/  =  -19.  '  l9Jf-6iV^=57. 

16aj-3?/  =  10.  '  l4p  +  6^  =  +  7. 

g     |8i>  +  5g  =  5.  |6a;-102/  =  5. 

I3p-2g  =  29.  '  ll5.v-14a;  =  -15. 

„     {5m-12n  =  -31.  ^^  |9c  +  8d5  =  -6. 

l3m  +  22w  =  -4.  *  [12c  +  10d  =  -7. 

g     |2a^-32/  =  -14.  ^g  f3e  +  7/=-23. 

l3aj  +  7^  =  48.  '  t5e  +  4/=-23. 

^     |5a.-  +  92/  =  8.  ^g_  J7^  +  8A:  =  -10. 


[6  2/ -9a;  =  -7.  ill  ^  + 6  A;  =  - 19. 

{7r-6f  =  63.  |5r-8.s  =  60. 

I9r  +  2^  =  13.  '    i6r  +  7s  =  -ll. 

169.    As  a  rule,  the  equations  containing  two  variables  do 
not  occur  in  as  simple  form  as  those  given  in  §§  167,  168. 

Example.     Solve  the  equations 

\— ^=0.  (1) 

[x(y-2)-y(x-5)=-13.  (2) 


SIMULTANEOUS   LINEAR   EQUATIONS 


227 


Solution  :  1.   Simplify  equation  (1) 
7  3 


=  0. 


x-\-S 

y  +  4 

.-.  7(y  +  4)- 

-Six 

+  3)  = 

=  0. 

3x 

-7y  = 

=  19. 

2. 

Simplify  equation  (2) : 

x(y-2)~ 

-y(x 

-5)  = 

:-13. 

2x 

-6y  = 

=  +13. 

(8) 


(4) 

3.   The  equations  (1)  and  (2)  thus  become  in  (3)  and  (4) : 

3x-7y  =  19.  (3) 

2a;-5y  =  13.  (4) 

Solve  these  equations  by  either  of  the  two  methods  of  elimination  ;  the 
solution  will  be  found  to  be  0-  =  4,  y  =—  1. 

Check  :  Substitute  the  values  of  x  and  y  in  equations  (1)  and  (2). 

Equations  (3)  and  (4)  are  called  the  Standard  Form  of  equations  (1) 
and  (2). 

Rule.  —  To  solve  complicated  simultaneous  linear  equations  hav- 
ing two  variables : 

1.  Reduce  the  equations  to  the  standard  form  by  clearing  of  frac- 
tions and  simplifying. 

2.  Solve  the  resulting  equations  by  either  of  the  two  methods. 

3.  Check  by  substituting  in  the  original  equations. 


EXERCISE  103 
Reduce  to  the  standard  form  and  solve : 


\2x 
3 

-'-i 

=  - 

-7 
2 

X 

A 

5 

11 
2 

• 

\Sp 

+  7g 

= 

12. 

1. 


2.    \p±2q 


+ 


^=1. 


4. 


10  m 


8  2/ 


11. 


m  +  3 


=  -17. 


3     2 

3  +  2^     l-j-5s 


5 


11 


=  4-2. 


228. 


ALGEBRA 


5. 


6. 


7. 


8.    i 


3   "^  4       6 


5^-8?^  = 


22 


9.    1 


'      +..^  =  0. 


m  -  2     2n-3 

1       +,,^=0. 


.2  7/1  +  5     3n-7 


r  +  ^      r  —  ^ 


2  3 


10. 


3      ■      4 

c  +  cZ-2^      1 
c  ~d  3 

3c  +  d-3 


re  +  a  -  5 


1-  a- 6 

2a-|-36  = 


2d-G 


11 


{a?  4-  5  y      2  y  +  a; 
13  11 

3x~7j  =  2. 


'     +^.  =  0. 


i»  —  1      2/  —  1 

5  7 


12. 


=  0. 


.2  x  -  3     2  y  +  13 

|(m  +  l)(7 
'    12  m  + 3^1 +  9  =  0, 
r  —  s     25      r  +  s 


2/      a;_„ 
3"2-^- 


2x     4  +  5?/ 


11 


13.    J  ('''  +  ^^  ("^  +  ^)  ~  ^'"^  +  ^)  (^  -  '^)  =112. 


14.    < 


15. 


2  6  3 

r  +  s  —  9      s  —  r  —  6 


=  0. 


w  —  2 _  10 -i(? _ p-10 ^ Q 
5  3  4      ~    ' 


16. 


17. 


j)  +  2      2w+p     ID  +  13 
6  32  16 

x-y    2^+y^Q 

3  2 

a;  +  2?/_a;__ll 
2  4~~T* 

4a  +  &      6a-36 


0. 


5  3 

8a  +  5&     10  a -6 


=  -3. 
=  -4. 


SIMULTANEOUS  LINEAR  EQUATIONS 


229 


170.  Certain  equations  in  which  the  variables  occur  in  the 
denominators  of  fractious  may  be  solved  readily  without 
clearing  the  equations  of  fractions. 


Example.     Solve  the  equations  ' 

X       y 

Vx      y 

(1) 

(2) 

Solution  :  Eliminate  the  term  containing  y : 

1.   Ms  (1):                              ^_  1^  =  40. 
X       y 

(3) 

2.   M,(2):                              24^46^_3 
X       y 

(4) 

3.  Add  (4)  and  (3):                     ^  =  37. 

(6) 

.-.  74  =  37x  or  x  =  2. 

(6) 

4.   Substitute  2  for  X  in  (1):  6 --  =  8. 

y 

(7) 

5.    Solve  (7)  fory:                           y=-3. 

Check  :  Substitute  x  =  2,  y  =  -  3  in  equations  (1)  and 

(2). 

EXERCISE  104 
Solve  the  following  sets  of  equations : 


1. 


2. 


a 

10  _ 

b 

:-l. 

5+ 

a 

15  _ 
h 

=  +  1. 

10 
c 

9 
d 

:4. 

8_ 
c 

l'>_ 
d 

9 
2* 

=-11. 

21 

=  — . 

2 

fi+i= 

5. 

iC 

y 

1 

1^ 

1. 

—  — 

.x 

y 

3_ 

i_ 

9. 

r 

« 

4  +  3^ 

-1. 

Vr 

s 

6 

,  4 

4 

+  - 

w 

v 

"5* 

9 

.  5 

_  7 

+  - 

.to 

i; 

~10 

230 


7.   < 


8.    < 


ALGEBRA 

'2      3          . 

A     B 

5      6_     1 
.A     B         2* 

9.    ^ 

5       7_29 

3x     y      9  ' 

3       5           9 

[x^Ay         8* 

'^-5  =  17. 

r      t 

5      6         3 
.r^t         2* 

10.    - 

r  5      4     1 

2  a;     3  2/     2 

2         17 

13  a;     2y~72 

171.  In  solving  problems  where  two  or  more  letters  are 
used  to  represent  unknown  numbers,  as  many  independent 
equations  must  be  obtained  from  the  conditions  of  the  problem 
as  there  are  letters  used. 

Example  1.  Four  seventeenths  of  the  greater  of  two  num- 
bers exceeds  the  less  number  by  5 ;  if  the  greater  be  divided 
by  the  less,  the  quotient  is  5  and  the  remainder  10.  Find  the 
two  numbers. 

Solution  :  1.     Let  g  =  the  greater  number, 

and  I  =  the  less  number. 

2.  Then,  ^\g  =  l+^.  (1) 

3.  When  the  greater  is  divided  by  the  less,  the  quotient  is  5  and  the 
remainder  10 ;  therefore, 

^  =  5  Z  +  10.  (2) 

4.  Solving  equations  (1)  and  (2),  g  =  85,  I  =  15. 
Check  :  j\  of  85  =  20  ;  20  -  15  =  5, 

and  85  =  5  •  15  +  10. 

Note.  In  equation  (2) ,  use  is  made  of  the  fact  that  the  dividend  equals  the 
divisor  times  the  quotient  plus  the  remainder.     Thus,  when  17  is  divided  by  2, 

17  =  8-2+1. 

Example  2.  If  3  be  added  to  both  numerator  and  denomi- 
nator of  a  fraction,  its  value  becomes  f ;  and  if  2  be  subtracted 
from  both  numerator  and  denominator  of  the  fraction  its  value 
becomes  ^.     Kequired  the  fraction. 


SIMULTANEOUS  LINEAR  EQUATIONS  231 

Solution  :  1.     Let  n  =  the  numerator, 

and  d  =  the  denominator. 

/.  -  =  the  fraction. 
d 

2.  By  the  first  condition  :  !i±l  =  ^.  (1) 

^  d  +  3     3  ^  ^ 

By  the  second  condition :         ^  ~     =  -  •  (2) 

d  —  2     2 

3.  Solving  the  equations  (1)  and  (2),  «  =  7,  d  =  12. 

Therefore  the  fraction  is  /tj. 

Check:  •        l+±  =  }^  =^- ■  J^l  =  l  =1. 

12 +J^      15      3     12-2      10      2 

Note.  Check  the  solution  by  going  back  to  the  conditions  of  the  problem. 


EXERCISE  105 

1.  Divide  59  into  two  parts  such  that  two  thirds  of  the  less 
shall  be  less  by  4  than  four  sevenths  of  the  greater. 

2.  Find  two  numbers  such  that  two  fifths  of  the  greater  ex^ 
ceeds  one  half  of  the  less  by  2,  and  four  thirds  of  the  less 
exceeds  three  fourths  of  the  greater  by  1. 

3.  If  5  be  added  to  the  numerator  of  a  certain  fraction,  the 
value  of  the  fraction  becomes  f ;  and  if  6  be  subtracted  from 
its  denominator,  the  value  of  the  fraction  becomes  f .  Find 
the  fraction. 

4.  If  9  be  added  to  both  terms  of  a  fraction,  its  value  be- 
comes 4 ;  and  if  7  be  subtracted  from  both  terms  of  the  frac- 
tion, its  value  becomes  |.     Find  the  fraction. 

5.  In  1910,  the  cost  of  3  tons  of  anthracite  coal  in  Phila- 
delphia exceeded  the  cost  of  4  tons  of  bituminous  coal  in  Bal- 
timore by  $3.10 ;  and  the  cost  of  9  tons  of  the  bituminous  coal 
exceeded  the  cost  of  5  tons  of  the  anthracite  by  90^.  Find 
the  cost  of  the  anthracite  and  of  the  bituminous  coal  in  1910. 

6.  A's  age  is  three  fifths  of  B's  age  ;  but  in  16  years  A's  age 
will  be  five  sevenths  of  B's  age.     Find  their  ages  at  present. 


232  ALGEBRA 

7.  If  twice  the  greater  of  two  numbers  be  divided  by  the 
less,  the  quotient  is  3  and  the  remainder  is  7 ;  if  five  times  the 
less  be  divided  by  the  greater,  the  quotient  is  2  and  the  re- 
mainder is  23.     Find  the  numbers. 

8.  On  the  tower  of  the  City  Hall  of  Philadelphia  is  a 
statue  of  William  Penn.  If  the  total  height  of  the  tower  and 
statue  be  divided  by  the  height  of  the  statue,  the  quotient  is  14 
and  the  remainder  is  29 ;  the  height  of  the  statue  exceeds  J^  of 
the  height  of  the  tower  by  7  feet.  Find  the  height  of  the 
tower  and  of  the  statue. 

9.  If  the  numerator  of  a  fraction  be  trebled,  and  the  denom- 
inator be  increased  by  8,  the  value  of  the  fraction  becomes  f ; 
and  if  the  denominator  be  halved,  and  the  numerator  be  de- 
creased by  7,  the  fraction  becomes  \.     Find  the  fraction. 

10.  The  City  Hall  of  Philadelphia  is  said  to  cover  a  greater 
area  than  any  other  building  in  the  United  States.  One  fifth 
of  its  width  exceeds  one  sixth  of  its  length  by  13  feet;  and 
one  ninth  of  its  length  exceeds  one  tenth  of  its  width  by  7  feet. 
Find  its  dimensions. 

11.  The  perimeter  of  a  certain  isosceles  triangle  (see  p.  102) 
is  140  inches.  The  side  exceeds  the  base  by  10  inches.  Find 
the  three  sides  of  the  triangle. 

12.  Three  years  ago  A's  age  was  -|  of  B's  age ;  but  in  nine 
years  his  age  will  be  ^-  of  B's  age.     Find  their  present  ages. 

13.  If  the  age  of  a  university  is  reckoned  from  the  date  of 
its  founding,  then,  in  1912,  the  age  of  Yale  exceeded  the  age  of 
Princeton  by  46  years.  Four  years  later,  one  half  of  the  age  of 
Princeton  will  exceed  one  third  of  the  age  of  Yale  by  13  years. 
Find  when  each  was  founded. 

14.  Twice  the  shorter  side  of  a  parallelogram  exceeds  the 
longer  side  by  5  inches ;  one  third  of  the  sum  of  the  shorter 
side  and  9  exceeds  one  fifth  of  the  longer  side  by  3.  Find  the 
sides  of  the  parallelogram. 


SIMULTANEOUS  LINEAR  EQUATIONS  233 

15.  In  1912  the  sum  of  the  ages  of  the  oldest  English  uni- 
versity, Oxford,  and  the  oldest  American  university.  Harvard, 
was  1316  years ;  the  age  of  Oxford  exceeded  three  and  three 
fourths  times  the  age  of  Harvard  by  5  years.  Find  when  each 
was  founded. 

16.  If  one  weight  is  placed  8  inches  from  the  fulcrum  of  a 
lever,  it  balances  another  weight  which  is  6  inches  from  the 
fulcrum  ;  if  the  first  weight  be  decreased  by  3  pounds,  and  the 
second  be  increased  by  4  pounds,  the  resulting  weights  will 
balance  if  placed  10  feet  and  5  feet  respectively  from  the  ful- 
crum.    Find  the  two  weights  (§  143). 

17.  The  sum  of  the  reciprocals  of  two  numbers  is  ^.  Twice 
the  reciprocal  of  the  greater  number  exceeds  the  reciprocal  of 
the  less  number  by  ^.     Find  the  two  numbers. 


j  Hint.    The  reciprocal  of  3  is  - ;   of  a  is  -  .  j 


18.  The  sum  of  the  reciprocals  of  two  numbers  is  ^.  If 
twice  the  reciprocal  of  the  less  be  increased  by  four  times  the 
reciprocal  of  the  greater,  the  sum  is  8.     Find  the  two  numbers. 

19.  A  purse  contained  $  6.55  in  quarters  and  dimes ;  after  6 
quarters  and  8  dimes  had  been  taken  out,  the  number  of 
quarters  equalled  three  times  the  number  of  dimes.  How 
many  of  each  kind  of  coin  were  there  ? 

20.  In  7  years,  A  will  be  three  times  as  old  as  B,  and  8 
years  ago  he  was  6  times  as  old.     What  are  their  present  ages  ? 

21.  The  length  of  a  room  exceeds  its  width  by  4  feet.  If  2 
feet  are  added  to  the  length,  and  3  feet  to'  the  width,  the  area 
is  increased  by  103  square  feet.     Find  the  dimensions. 

Solution:  1.      Let  Z  =  the  number  of  feet  in  the  length, 
and  w  =  the  number  of  feet  in  the  width. 
/.  he  =  the  area. 
2.  l  =  w-\-^.  (1) 


234  ALGEBRA 

3.  l-\-2  =  the  new  length. 
w  -\-S  =  the  new  width. 

.'.  (I  +  2)  (w  +  S)  =  the  new  area. 

4.  .-.(«+  2)  (MJ  4-  3)  =  ho  +  103.  .  (2) 

5.  Solve  the  equations  (1)  and  (2),  and  check. 

22.  If  a  rectangular  lot  were  6  feet  longer  and  5  feet 
wider  than  it  is  now,  it  would  contain  839  square  feet  more ; 
if  it  were  4  feet  longer,  and  7  feet  wider,  it  would  contain  879 
square  feet  more.     Find  its  length  and  width. 

23.  If  one  weight,  increased  by  10  pounds,  be  placed  6  feet 
from  the  fulcrum,  it  will  balance  a  second  weight,  placed  4i 
feet  from  the  fulcrum ;  if  the  second  weight,  increased  by  10 
pounds,  be  placed  3^  feet  from  the  fulcrum,,  it  will  balance  the 
first  weight  placed  6  feet  from  the  fulcrum.  Find  the  two 
weights. 

24.  Will  weighs  50  pounds.  When  Will  seats  himself  6 
feet  from  the  fulcrum  and  John  seats  himself  4  feet  from  the 
fulcrum  on  the  same  side,  they  exactly  balance  James,  who  is 
sitting  7\  feet  from  the  fulcrum  on  the  other  side.  When 
Will  and  John  change  places,  they  find  that  James  must  sit  7| 
feet  from  the  fulcrum.  How  much  do  John  and  James 
weigh  ? 

25.  A  crew  can  row  10  miles  down  stream  in  50  minutes, 
and  12  miles  up  stream  in  an  hour  and  a  half.  Find  the  rate 
in  miles  an  hour  of  the  current,  and  of  the  crew  in  still  water. 

(Hint.  Review  Biver  Problems,  Exercise  90,  §  144  ;  express  50  min- 
uted as  f  hour. ) 

26.  A  motor  boat  which  can  run  at  the  rate  of  15  miles  an 
hour  in  still  water  went  downstream  a  certain  distance  in  4 
hours.  It  took  6  hours  for  the  trip  back.  What  was  the  dis- 
tance and  the  rate  of  the  current  ? 

27.  A  crew  are  rowing  on  a  stream  the  rate  of  whose  current 
is  known  to  be  2  miles  an  hour;  they  find  that  it  takes  them 


SIMULTANEOUS  LINEAR  EQUATIONS  235 

one  and  one  third  hours  to  go  down,  and  four  hours  to  come 
back  a  certain  distance.  Find  the  distance  and  the  rate  of  the 
crew  in  still  water. 

28.  An  express  train  travels  30  miles  in  27  minutes  less 
time  than  a  slow  train.  If  the  rate  of  the  express  train  were 
J  as  great,  and  if  the  rate  of  the  slow  train  were  ^  as  great, 
the  express  train  would  travel  30  miles  in  54  minutes  less 
time  than  the  slow  train.  Find  the  rate  of  each  train  in  miles 
an  hour. 

29.  If  a  field  is  made  5  feet  longer  and  7  feet  wider,  its  area 
would  be  increased  by  830  square  feet;  but  if  its  length  is 
made  8  feet  less,  and  its  width  4  feet  less,  its  area  is  dimin- 
ished by  700  square  feet.     Find  its  length  and  width. 

30.  The  fore  wheel  of  a  carriage  makes  8  revolutions  more 
than  the  hind  wheel  in  going  180  feet ;  but  if  the  circumfer- 
ence of  the  fore  wheel  were  |  as  great,  and  of  the  hind  wheel 
f  as  great,  the  fore  wheel  would  make  only  5  revolutions  more 
than  the  hind  wheel  in  going  the  same  distance.  Find  the 
circumference  of  each  wheel. 

31.  A  man  has  $  2500  invested  from  which  he  receives  a 
total  income  of  $  135.  Part  of  the  money  is  invested  at  6  % 
and  part  at  4^  %.     How  much  is  invested  in  each  way  ? 

32.  A  man  has  altogether  $5000  of  savings.  He  has  part 
invested  in  a  5  %  bond,  and  the  balance  invested  in  a  mort- 
gage drawing  6%.  If  his  total  income  is  $280,  how  much 
has  he  invested  in  each  way  ? 

33.  A  man  has  $1200  invested  at  one  rate  of  interest  and 
$  500  at  a  rate  which  is  1  per  cent  greater  than  the  former 
rate.  The  income  from  the  first  investment  exceeds  the  in- 
come from  the  second  investment  by  $23.  Find  the  rate  at 
which  each  sum  is  invested. 

34.  The  simple  interest  on  $800  at  5  %  for  a  certain  num- 
ber of  years  exceeds  the  simple  interest  on  $300  at  6  %  for  a 


236  ALCxEBRA 

second  period  of  years  by  $  60.  If  the  second  period  of  years 
exceeds  the  first  by  4  years,  find  the  number  of  years  each  sum 
is  invested.    ' 

35.  There  are  two  supplementary  angles  such  that  ^  of  the 
larger  exceeds  -f  of  the  smaller  by  5°.     Find  the  angles.     (§  13.) 

36.  One  angle  of  a  triangle  is  35°.  If  the  number  of  degrees 
in  one  of  the  remaining  two  angles  is  divided  by  the  number 
in  the  other,  the  quotient  is  9  and  the  remainder  is  10.  Find 
the  three  angles  of  the  triangle.     (See  §  13.) 

37.  The  fastest  train  on  the  Pennsylvania  R.  E,.  makes  the 
trip  between  Chicago  and  Fort  Wayne,  Indiana,  a  distance  of 
148  miles,  in  1  hour  and  20  minutes  less  time  than  one  of  the 
ordinary  trains.  Its  rate  is  fthat  of  the  ordinary  train.  Find 
the  rate  of  each  train. 

38.  A  and  B  working  together  can  do  a  piece  of  work  in  6 
days ;  they  can  also  complete  the  work  if  A  works  10  days 
and  B  3  days.  How  many  days  would  it  take  each  of  them  to 
do  the  work  alone  ?     (See  §  142.) 

39.  A  and  B  working  together  can  do  a  piece  of  work  in  7^ 
days.  If  A  works  alone  for  3  days,  and  B  alone  for  6  days, 
they  would  complete  -j\  of  the  work.  Find  how  long  it  would 
take  each  alone  to  do  the  work. 

40.  A  man  has  a  sum  of  money  invested  at  a  certain  rate  of 
interest.  Another  man  has  a  sum  greater  by  $  3000,  invested 
at  a  rate  1  %  less,  and  his  income  is  $45  less  than  that  of  the 
first.  A  third  man  has  a  sum  less  by  $2000  than  that  of  the 
first,  invested  at  a  rate  1  %  greater,  and  his  income  is  $  40 
greater  than  that  of  the  first.  Find  the  capital  of  each  man, 
and  the  rate  at  which  it  is  invested. 

172.  Relations  between  the  Digits  of  a  Number.  Integral 
numbers  are  written  by  means  of  the  digits. 

Thus,  372  is  a  number  of  three  digits;  3  represents  300 
units,  7  represents  70  units,  and  2  represents  2  units. 


SIMULTANEOUS  LINEAR  EQUATIONS  237 

Similarly,  if  t  is  the  tens^  digit  and  u  is  the  units'  dlrjit,  the 
number  is  10  t  4-  u. 

When  the  digits  of  a  number  are  reversed,  a  new  number  is 
formed;  thus,  reversing  52  gives  25.     Notice  that 

52  =  10  X  5  4-  2  and  25  =  10  X  2  -f-  5. 

Similarl}^,  if  x  and  y  are  the  tens  and  units  digits  of  a 
number,  the  number  is  10  x-\-  y,  if  the  digits  are  reversed,  the 
new  tens'  digit  is  ?/,  the  new  units'  digit  is  it',  and  the  new 
number  is  10  y  +  x. 

EXERCISE  106 

1.  Write  the  number  whose  units'  digit  is  a,  tens'  digit  6, 
and  hundreds'  digit  c. 

2.  Write  the  number  represented  by  reversing  the  digits 
in  Example  1. 

3.  Representing  the  tens'  digit  of  a  number  by  t,  and  the 
units'  digit  by  u  represent: 

(a)  the  sum  of  the  digits ; 
(6)  the  number ; 

(c)  the  product  of  the  digits; 

(d)  the  quotient  when  the  number  is  divided  by  the  sum  of 
the  digits. 

4.  Representing  the  tens'  digit  of  a  number  by  x,  and  the 
units'  digit  by  y. 

(a)  express  the  original  number; 

(b)  express  the  number  obtained  by  reversing  the  digits; 

(c)  express  the  quotient  of  the  new  number  divided  by  the 
old  number; 

(d)  express  by  an  equation  the  fact  that  when  the  original 
number  is  divided  by  the  sum  of  its  digits  the  quotient  is  5 
and  the  remainder  is  3.     (See  note,  Example  1,  §  171.) 


238  ALGEBRA 

5.  The  sum  of  the  two  digits  of  a  number  of  two  digits  is 
11 ;  if  the  digits  be  reversed,  the  quotient  of  the  new  number 
divided  by  the  old  is  2  and  th>e.  remainder  is  7.  Find  the 
number. 

Solution  :  1.     Let  t  =  the  lens'  digit, 

and  u  =the  units'  digit. 

.■.t  +  u  =  IL  (1) 

2.  lot  -\-  u=  the  original  number, 
and  \0u  +  t  =  the  new  number. 

.-.  10«  +  «=:  2(10^4- w)+ 7.  (2) 

3.  Solving  the  pair  of  equations,  (1)  and  (2),  gives  i  =  3,  w  =  8. 
.*.  the  number  is  38. 

Check  :  The  sum  of  the  digits  is  11. 

83  -~  38  gives  the  quotient  2  and  the  remainder  7. 

6.  The  tens'  digit  of  a  number  exceeds  its  units'  digit  by 
4.  If  the  digits  be  reversed,  the  new  number  is  6  more  than 
one  half  of  the  old  number.     Find  the  number. 

7.  The  sum  of  the  two  digits  of  a  number  is  9.  If  the 
digits  be  reversed,  the  quotient  of  the  new  number  divided  by 
the  units'  digit  of  the  given  number  is  13  and  the  remainder 
is  1.     Find  the  number. 

8.  The  sum  of  the  two  digits  of  a  number  is  16 ;  and  if  18 
be  subtracted  from  the  number,  the  remainder  equals  the 
number  obtained  by  reversing  the  digits.     Find  the  number. 

9.  If  the  digits  of  a  number  of  two  figures  be  reversed,  the 
sum  of  the  resulting  number  and  twice  the  given  number  is 
204 ;  and  if  the  given  number  is  divided  by  the  sum  of  its 
digits,  the  quotient  is  7  and  the  remainder  is  6.  Find  the 
number. 

10.  If  a  certain  number  be  divided  by  the  sum  of  its  two 
digits,  the  quotient  is  4  and  the  remainder  is  3.  If  the  digits 
be  reversed,  the  sum  of  the  resulting  inimber  and  23  is  twice 
the  given  number.     Find  the  given  number. 


SIMULTANEOUS  LINEAR  EQUATIONS  239 

173.  Literal  Simultaneous  Equations.  In  solving  literal  si- 
multaneous equations,  the  addition  or  subtraction  method  of 
elimination  is  usually  the  best. 

Example.     Solve  for  x  and  y  the  equations : 


ax-^by  =  c. 

(1) 

rx-{-  sy  =  t. 
Solution  :  Eliminate  y. 

(^) 

1.   M,  (1)  :                                          sax  -\-  shy  =  sc. 

(3) 

2.    Mft  (2)  :                                            brx  -f-  bsy  =  bt. 

(4) 

3.   Subtract  (4)  from  (3)  :                sax  —  brx  =sc  — 

bt. 

(5) 

.'.  {sa  —  br)x  =  (sc  - 

-bt). 

-hn 

-br) 

(0) 

Now,  going  back  to  (1)  and  (2),  eliminate  x. 

4.   Mr  (1):                                          rax  +  rby  =  re. 

(7) 

5.   Ma  (2) :                                         rax  +  say  =  ta. 

(8) 

6.    Subtract  (8)  from  (7) :               (rb  -  sa)y  =  (re  - 

-ta). 

^      (rb- 

-  ta)^ 
-sa) 

The  solution  is  :                      x  =  ^^  ~  ^S    2/  =  ^ 

ta     ta  —  re 

sa  —br  rb  —  sa     sa  —  br 

Note.  In  step  6,  when  subtracting  say  from  rby,  we  notice  that  these 
are  like  terms  since  they  have  the  cominon  factor  y,  and  then  subtract  by 
multiplying  that  common  factor  y  by  the  difference  of  its  coefficients, 
(rb  —  sa).  Or,  we  may  think  of  the  difference  as  being  rby  —  say,  which, 
factored,  becomes  (rb  —  sa)y. 

EXERCISE  107 

\Sx  +  4:y  =  7a.  {2ax-\-.y  =  b. 

[2 x  —  5y  =  6b.  '    lax—  2y=  c. 

{Sx-\-ay  =  5.  imx-^ny=p. 

'    [2  x  —  by  =  6.  '    \cx-^dy  =  e. 

{4:X-\-my  =  n.  (ax-{'by  =  a^  +  Sab. 

\x-{-py  =  q,  '    \2ax  — 3 by =2a^  — 4: ab. 


240 


ALGEBRA 


7. 


8. 


X  +  ay  =  ab  —  2  a. 
hx-{-ay  =  —  ah. 

{ax— by  =  2  ah. 
[2hx-\-2ay^3h^-a^. 

hx-\-ay  =  a^  +  ah. 
x  —  y  =  a-{-h. 

2cx  +  dy  =  2c'-\-d\ 


10.    \x  + 


=  3. 


11. 


12. 


13. 


9Lj^'^^a''  +  h\ 
x     y 

ah        2      1,2 
X     y 

rx  ■}-sy  =  0. 
sx-\-ry  =  s^—  r^o 

mx  —  ny  =  0. 


x-y  = 


myi 


14. 


x-\-ay  =  h. 
hx-\-y  =  a. 


15. 

is  n. 


Find  two  numbers  whose  sum  is  m  and  whose  difference 


16.  Divide  the  number  c  into  two  parts  so  that  when  the 
larger  is  divided  by  the  smaller  the  quotient  is  d. 

17.  Divide  the  number  ?-  into  two  parts  such  that  when  the 
larger  is  divided  by  the  smaller  the  quotient  is  s  and  the  re- 
mainder is  t. 

18.  A  and  B  can  do  a  piece  of  work  together  in  k  days ;  if 
A  works  3  days  and  B  works  5  days  they  can  do  -i-  of  it.  Find 
how  long  it  would  take  each  alone  to  do  the  piece  of  work. 

19.  The  sum  of  the  digits  of  a  number  of  two  digits  is  a; 
the  number  itself  equals  b  times  the  units  digit.  Find  the 
digits  of  the  number. 

20.  If  a  be  added  to  the  numerator  and  b  be  added  to  the  de- 
nominator of  a  certain  fraction,  its  value  becomes  1 ;  if  h  be 
added  to  the  numerator  and  a  to  the  denominator,  its  value  be- 
comes 2.     Find  the  fraction. 

21.  Find  two  numbers  whose  sum  is  c  and  such  that  h  times 
the  first  exceeds  a  times  the  second  by  d. 

22.  Find  two  numbers  such  that  the  quotient  of  a  divided 
by  the  greater  exceeds  by  c  the  quotient  of  b  divided  by  the 


SIMULTANEOUS  LINEAR  EQUATIONS  241 

less ;  and  such  tliat  the  quotient  of  h  divided  by  the  greater 
exceeds  by  d  the  quotient  of  a  divided  by  the  less. 

The  following  two  sets  of  equations  arise  in  the  solution  of 
problems  in  applied  mathematics;  find  Tand/: 

23.    I ^^  -  ^  =  '^^/-  24.    )  ^"^  ~^^  ^*•^• 


T=vf.  [T-ng  =  nf. 

EQUATIONS  CONTAINING  THREE  VARIABLES 

174.  In  the  preceding  paragraphs  equations  having  two 
variables  have  been  solved;  in  each  case  two  equations  were 
given.  It  is  interesting  to  study  equations  with  more  than 
two  variables.  For  three  variables,  three  equations  are 
necessary. 

Example.     Solve  the  set  of  equations, 

\2  x  —  y  -\-z  =  5.  (1) 

\Sx  +  2y-{-3z  =  7.  (2) 

l4a;_3y-5^  =  -3.  (3) 

Solution  :  Eliminate  z  by  combining  (1)  and  (2)  ;  the  resulting  equa- 
tion will  contain  only  x  and  y. 

1.  Ma  (1)  :  6x-Sy-\-Sz  =  lb.  (4) 

2.  Subtract  (2)  from  (4)  :  3  a;  -  5  ?/  =  8.  (5) 
Now. eliminate  z  by  combining  (1;  and  (3);  the  resulting  equation  will 

contain  only  x  and  y. 

3.  M5(l):  10 x-5?/ +52  =  25.  (6) 

4.  Add  (3)  and  (6)  :  14  x -  8  ?/  =  22.  (7) 

This  gives  the  equations  : 

3x-6y=S.  (5) 

Ux-Sy  =  22.  (7) 

Solve  this  set  of  equations  for  x  and  y. 
The  solution  is,  x  =  1,     y  =—  \. 
Substitute  these  values  of  x  and  y  in  (1)  and  obtain  z. 

2  +  1  +  2;  =  5. 
.-.  z  =  2. 
The  complete  solution  is  .r  =  1,    y  =  —  1,    z  =  2. 
Check  the  solution  by  substituting  it  in  each  of  the  given  equations. 


242 


ALGEBRA 


EXERCISE   108 

a  -  2  6  +  c  =  0. 
a-b-{-2c  =  -ll. 
2a-6  +  c  =  -9. 


{Sx~2y  =  l. 

2.  30^4-4^  =  5. 
[5z-\-3y  =  4:. 

{12  m  — 4:71 -{-p=  3. 

3.  Un  —  71  —  2  p  =  —  1. 
[5m  —  271  =  0. 

i5r-3s-4t  =  -l. 

4.  2s  +  3^  =  H-9. 
[4r-^  =  3. 

pa^-f  4r+2«  =  -5. 

5.  2x-3r-4^  =  -10. 
[4a;  +  2r  +  3^  =  -21. 

(6A-2B-3C  =  3. 

6.  2^  +  35  +  5(7=0. 
[8^-55-60=1. 


7. 


8. 


10. 


2      1  ,  1      f, 

+  -  =  5. 

X      y      z 

111^ 

-  +  -  +  -  =  9. 

a;      1/      2; 

^-^  =  -2. 

iC         2 

11,1      1 

-+-  +  -  =  ;:• 

X     y      z     3 

1_1_1__5 

X     y      z         3 

1_1_1^25 

y      z      X       3 

x-\-y-\-2z  =  a. 

2x-y  +  z=:b. 

x  —  2y±z  =  c. 

1_1_1_^ 

X     y      z        ' 

1-1-1  =  6. 

y      z      X 

111 

=  c. 

z      X     y 

11.  The  angle  .^  of  a  triangle  exceeds  the  angle  B  by  20°; 
and  the  angle  C  exceeds  the  angle  A  by  20°.  Find  the  three 
angles  of  the  triangle.     (See  §  13.) 

12.  The  perimeter  of  a  triangle  is  175  inches.  The  side  a  is 
20  inches  less  than  twice  the  side  6;  and  the  side  h  exceeds 
the  side  c  by  5  inches.     Find  the  sides  of  the  triangle. 

13.  The  sum  of  the  sides  a  and  6  of  a  triangle  is  147  inches ; 
the  sum  of  the  sides  h  and  c  is  135  inches ;  and  the  perimeter 
is  219  inches.     Find  the  three  sides  of  the  triangle. 


SIMULTANEOUS  LINEAR   EQUATIONS  243 

14.  The  total  area  of  the  three  largest  oceans  is  134  million 
square  miles.  The  area  of  the  Pacific  Ocean  exceeds  twice  the 
area  of  the  Atlantic  Ocean  by  1  million  square  miles;  and  the 
area  of  the  Atlantic  Ocean  exceeds  the  area  of  the  Indian 
Ocean  by  7  million  square  miles.  Find  the  area  of  each  of  the 
oceans. 

15.  A  recipe  for  a  fondant  for  candy  calls  for  a  total  of 
three  and  five  sixths  cups  of  sugar,  water,  and  glucose ;  twice 
the  total  amount  of  water  and  glucose  exceeds  the  amount  of 
sugar  by  one  sixth  cup ;  and  three  times  the  amount  of  glucose 
and  twice  the  amount  of  sugar  make  six  cups.  How  many 
cups  of  each  ingredient  is  required  for  the  fondant  ? 

16.  The  sum  of  the  three  digits  of  a  number  is  13.  If  the 
number,  decreased  by  8,  be  divided  by  the  sum  of  its  units 
and  tens'  digits,  the  quotient  is  25 ;  and  if  99  be  added  to  the 
number,  the  digits  will  be  reversed.     Find  the  number. 

Solution  :  1.    Let  x  =  the  hundreds'  digit, 

y  =  the  tens'  digit, 
and  z  =  the  units'  digit. 

2.  Then        100  r  -\- \0  y  -]-  z  =  the  number, 

3.  and  100  2  +  10  y  +  x  =  the  number  with  its  digits  reversed. 

4.  By  the  conditions  of  the  problem, 

x  +  y  +  z  =  l'S, 
lOOx+lOy  +  g  —  8  _25 

y-h  z 

and  100x-\-10y  +  z  +  99  =  l00z-\-l0y  +  x. 

5.  Solving  these  questions,  x  =  2,  y  =  S,  z  =S. 
Therefore  the  number  is  283. 

17.  The  sum  of  the  three  digits  of  a  number  is  23 ;  and  the 
digit  in  the  tens'  place  exceeds  that  in  the  units'  place  by  3. 
If  198  be  subtracted  from  the  number,  the  digits  will  be 
reversed.     Find  the  number. 

18.  A  and  B  can  do  a  piece  of  work  in  10  days,  A  and  C  in 
12  days,  and  B  and  C  in  20  days.  In  how  many  days  can  each 
of  them  alone  do  it  ? 


XIV.     SQUARE   ROOT  AND   QUADRATIC   SURDS 

175.  Square  Root  by  Inspection.  The  square  root  of  a  per- 
fect square  monomial  (§  90),  and  of  a  perfect  square  trinomial 
(§  96)  have  been  found  by  inspection.    Eeview  these  paragraphs. 

176.  Tv/o  Square  Roots  are  obtained  for  each  number.    They 
are  of  equal  absolute  value,  but  have  opposite  signs ;  they  may   , 
be  written  together,  by  means  of  the  double  sign,  ± . 

Example  1.     VOo^p" ^ _j_  3 ^sjr,  .  since  (+  3 gi%Y  =  9 a^^^  ; 
and  (- 3  a26)2:^  9^2452. 

Example  2.  VOa^-  12aa:  +  4x2  =  ±(3  a  -  2x)  ; 

since       {+(3  a- 25c)}2  =  +  (3  a  -  2ic)2  =  Qa^  -  12ax  +  4ic2, 
and         {_ (3  a  _  2  x)f  =  +  (3  a  -  2 x)2  =  9  a2  -  12  ax  +  4x2. 

177.  The  square  root  of  a  large  number  may  sometimes  be 
obtained  by  inspection  by  factoring  the  number. 

Example.     VlTeTa*  =  \/4  •  441  a*  =±2 -^Icfi  =±^2.  a^. 

EXERCISE  109 

Find  the  square  roots  of : 
.     ^o     4  6  ^      16ai«  ^     169  a^y 

2.  64a;V2^  '   4a^  ^     121  cH^ 

3.  144aW.  *    1^9^'*  *    144^2/^' 

^     196  m«  ,^     225  s«^^ 

4.  225 /m  7.    ^^^.  10.    ^^^. 

11.  When  is  a  trinomial  a  perfect  square? 
Find  the  value  of : 

12.  Va^-6a262  +  9  6^  14.    Vl69  a^  -  26  ar^  +  r«. 


13.    Vm«-10m3n  +  25n2.  15.    V4  aj^  -  20  a;?/^  +  25  2/\ 

244 


SQUARE  ROOT   AND   QUADRATIC   SURDS         245 


16.    V3136.  18.    V4225a262.  20.    V5184r«s». 


a^  +  2  a6  +  b'^ 

2a 

2fl 

+  2a6  +  63 
+  2a&  +  62 

17.    V2916a;y.         19.    V5625mV.  21.    Vll025ar?/V. 

178.  Square  Root  found  by  Long  Division.  If  it  is  not  pos- 
sible to  factor  readily  the  number  under  the  radical  sign,  the 
square  root,  if  there  is  one,  may  be  found  by  a  process  like 
long  division. 

Example  1.     Find  the  square  roots  of  a-  -f-  2  a6  +  b^. 

_  a+  6 

Solution  :  1.    s/a^  =  a.    Place  a  in  the  root. 

2.  Square  a  ;  subtract. 

3.  2  X  a  =  2  a.     Trial  divisor. 
2ab  -T-2a  =  b.    Add  b  to  the  trial  divisor  and 

to  the  root.     Complete  divisor. 

4.  &  X  (2  a  +  6)  ;  subtract. 

The  square  roots  are  :  +(«  +  &)  and  —(a-\-b). 

Explanation  :  1.  Find  the  square  root  of  the  first  teim,  obtaining  a, 
the  first  term  of  the  root ;  place  it  in  the  root. 

2.  Square  the  first  term  of  the  root  and  subtract  it  from  the  given 
number,  obtaining  the  first  remainder,  '2ab  +  b'\ 

3.  Double  the  first  term  of  the  root,  obtaining  2  a,  the  trial  divisor. 
Divide  the  first  term  of  the  remainder  by  2  «,  obtaining  6,  the  second  term 
of  the  root.  Add  b  to  the  root  and  to  the  trial  divisor ;  the  complete 
divisor  is  2  a  +  6. 

4.  Multiply  the  complete  divisor  by  b  and  subtract. 

Step  3  is  suggested  by  the  process  of  squaring  a  binomial.  When 
squaring  a  binomial,  the  middle  term  is  obtained  by  taking  twice  the 
product  of  the  fii-st  and  second  terms  ;  this  is  equivalent  to  taking  twice 
the  first  tenn  and  multiplying  by  the  second.  Reversing  the  process,  the 
second  term,  6,  will  be  found,  if  2  ab  is  divided  by  2  a.  After  a^  is  sub- 
tracted from  a^  -\-2ab  +  6^  the  remainder  2a6  +  b'^  equals  6(2  a  +  b). 
This  suggests  adding  b  to  the  trial  divisor  and  multiplying  the  sum  by  b. 

Example  2.    Eind  the  square  root  of 

20ar^-70a;  +  4a;*-h49-3ar^. 


246 


ALGEBRA 


Subtract. 

ix^+^Ox^-  8:r-^-70x+4d 
4x4 

4x' 

+5x 

20x^-  3x^-70x+49 

4a:H6x 

20x3  +  25x2 

4x2  +  10  a: 

-7 

-28x2-70x+4d 

7). 

4a;H10:« 

—  7 

-28x2-70x+49 

Solution  :  1.   Arrange  it  in  the  descending  powers  of  x : 
2x2  +  5x-7 

2.  V4  x4  =  2  x2. 

3.  (2x2)2=4x4. 

4.  2  X  (2  x2)  =:  4  x2. 
20  x3  -4-  4  x2  =  5  X. 

5.  5x(4x2  +  5x). 

6.  2x(2x2  +  5x). 

7.  -  28x2 --4x2=- 7- 

8.  _7(4x2  +  10x 

The  square  roots  are 

Rule.  —  To  find  the  square  root  of  an  algebraic  expression : 

1.  Arrange  it  according  to  ascending  or  descending  powers  of  some 
letter. 

2.  Write  the  positive  square  root  of  the  first  term  of  the  given  expres- 
sion as  the  first  term  of  the  root.  Square  it  and  subtract  the  result 
from  the  given  expression. 

3.  Double  the  root  already  found,  for  the  trial  divisor.  Divide  the 
first  term  of  the  remainder  by  the  first  term  of  the  trial  divisor.  Add 
the  quotient  to  the  root  and  also  to  the  trial  divisor,- obtaining  the  com- 
plete divisor. 

4.  Multiply  the  complete  divisor  by  the  new  term  in  the  square  root ; 
subtract  the  product  from  the  remainder  obtained  in  step  2. 

5.  Continue  in  this  manner  :  (a)  double  the  root  already  found  for  a 
new  trial  divisor  ;  (6)  divide  the  first  term  of  the  remainder  by  the  first 
term  of  this  product  for  the  new  term  of  the  root ;  (c)  add  the  new  term 
of  the  root  to  the  trial  divisor,  obtaining  the  complete  divisor  ;  (d)  multi- 
ply the  complete  divisor  by  the  new  term  of  the  root ;  (e)  subtract. 


EXERCISE   110 

Find  the  square  roots  of  the  following : 

2.  25m^-\-S0mn  +  9n\  5.    l-6a  +  lla^- 6a^ +  a\ 

3.  S6a^-12ab  +  b\  6.    9x'-24.a^-\-4.x' +Wx  +  4:. 

7.  49a2_30a3  +  16^9a*-40a. 

8.  25a''-20a^y-26x'y'-\-12xy  +9y*. 


SQUARE   ROOT   AND   QUADRATIC   SURDS  247 

9.  9a^  +  l-4a34.4a*'-Ga--f  12a^ 

10.  16  m*  +  8  ?/iV  -  23  m^x*  -  {jmx^  +  9  a-^. 

11.  l-2x-^3x^-4x^  +  ^x*-2x'^-^x^'. 

12.  a^  -  4  x'a'  +  10  a%''  +  4  ar^a^  -  20  xa'  +  25  a«. 

13.  9x^  +  25y^-\-16z^  +  30xy-24:Xz-40yz. 

14.  a2  +  62  4.c2-2a6-2ac  +  2^;c. 

15.  20  iib^  +  9  a*  -  26  a-?>2  +  25  6*  -  12  r/6. 

16.  20ar*'-70a-  +  4a-^  +  49-3ar^. 

17.  49  m*  —  14  mhi  —  55m^n^  +  8  mii^  + 16  n*. 

18.  7/i2  +  8m  +  12-  — +  4/ 

19.  ^     ^^        ^    +2a:^4a^ 
_     16  ,  8a-     13ar^     4ar'  ,  4a^ 

20.  -7r-\-- — -T^^ r  +  — r' 

9       6  a      6a^        (t        a* 

179.  Square  Root  of  an  Arithmetical  Number.  The  square 
root  of  100  is  10;  of  10,000  is  100;  6tc.  Hence  the  square 
root  of  a  number  between  1  and  100  is  between  1  and  10 ;  the 
square  root  of  a  number  between  100  and  10,000  is  between  10 
and  100 ;  etc. 

That  is,  the  integral  part  of  the  square  root  of  a  number  of 
one  or  two  figures  contains  one  figure ;  of  a  number  of  three  or 
four  figures,  contains  two  figures;  and  so  on. 

Hence,  if  the  given  number  is  divided  into  groups  of  two 
figures  each,  beginning  with  the  units  figure,  for  each  group 
in  the  number  there  will  be  one  figure  in  the  square  root.  The 
groups  are  called  Periods. 

Thus,  2346  becomes  23  45  ;  it  has  two  periods  and  its  square  root  has 
tw(5  'figures,  a  tens'  and  a  units'  figure. 

34038  becomes  3  40  38  ;  it  has  three  periods  and  its  square  root  has 
tbi'fee  figures.  A  number  having  an  odd  number  of  figures  will  always 
have  only  one  figure  in  its  left-hand  period,  as  in  this  case. 


248 


ALGEBRA 


A  decimal  number  is  divided  in  the  same  manner,  starting 
from  the  decimal  point  in  both  directions. 

Thus,  3257.846  becomes  32  57.84  60.  The  last  decimal  period  is  always 
completed  by  aniiexijig  a  zero.  This  number  has  two  figures  before  the 
decimal  point  and  two  after  it,  in  its  square  root. 


180.  The  first  figure  of  the  square  root  of  a  number  is  found 
by  inspection ;  the  remaining  figures  are  found  in  the  same 
manner  as  the  square  root  of  a  polynomial. 


Example  1.    Find  the  square  root  of  4624. 


There  are 


60  +  8 


46  24 
36  00 


Solution,     1.    Divide  4624  into  periods  ;  this  gives  46  24. 
in  the  square  root  a  tens'  and  a  units'  figure. 

2.  The  tens'  figure  must  be  6  ;  7  is  too  large  for  70^  =  4900,  which  is 
more  than  4624. 

3.  The  rest  of  the  square  root  is  found  as  follows  : 
3600  is  the  largest  square  less  than  4000. 
VSOOO  =  60  ;  place  60  in  the  root. 
Square  60  and  subtract. 

Double  60.     Trial  divisor.  120 

102  -=-  12  =  8+.     Place  8  in  root  and  add  to  trial  divisor.      __8 

Complete  divisor  128 

Multiply  complete  divisor  by  8. 
The  square  roots  are  -f  68  and  —  68. 

It  is  customary  to  abbreviate  the  solution  by  omitting  the 
zeros  as  in  the  following  example. 

Example  2.   Eind  the  square  root  of  552.25. 


10  24 
10  24 


Solution.     The  largest  square  less  than  5  is 
Place  2  in  the  root. 

2x2  =  4;  annex  0.     Trial  divisor. 

15-^4  =  3+;  add  3  to  the  trial  divisor. 

Complete  divisor.     Multiply  by  3. 

2  X  23  =  46  ;  annex  0.     Trial  divisor. 

230  -4-  46  =  6+ .     Add  5  to  the  trial  divLsor. 

Complete  divisor.     Multiply  by  5. 

The  square  roots  are  +  23.5  and  —  23.5. 


4  ;    V4  =  2. 


23.5 

5  52.26 

4 

40 

162 

3 

43 

129 

460 

23  26 

.  5 
466 

23  25 

SQUARE   ROOT  AND  QUADRATIC   SURDS         249 

Rule.  —  To  find  the  square  root  of  an  arithmetical  number  : 

1.  Separate  the  number  into  periods  (§  179). 

2.  Find  the  greatest  square  number  in  the  left-hand  period  ;  write  its 
positive  square  root  as  the  first  figure  of  the  root ;  subtract  the  square  of 
the  first  root  figure  from  the  left-hand  period,  and  to  the  result  annex  the 
next  period. 

3.  Form  the  trial  divisor  by  doubling  the  root  already  found  and  an- 
nexing zero. 

4.  Divide  the  remainder  by  the  trial  divisor,  omitting  the  last  figure 
of  each.  Annex  the  quotient  to  the  root  already  found ;  add  it  to  the 
trial  divisor  for  the  complete  divisor. 

5.  Multiply  the  complete  divisor  by  the  root  figure  last  obtained  and 
subtract  the  product  from  the  remainder. 

6.  If  other  periods  remain,  proceed  as  before,  repeating  steps  3,  4,  and 
5  until  there  is  no  remainder  or  until  the  desired  number  of  decimal 
places  has  been  obtained  for  the  root. 

Notp:  1.  It  sometimes  happens  that,  on  multiplying  a  complete  divisor 
by  the  figure  of  the  root  last  obtained,  the  product  is  greater  than  the 
remainder.  In  such  cases,  the  figure  of  the  root  last  obtained  is  too 
great,  and  the  next  smaller  integer  must  be  substituted  for  it. 

Note  2.  If  any  figure  of  the  root  is  0,  annex  0  to  the  trial  divisor  and 
annex  to  the  remainder  the  next  period. 

Example  3.   Find  the  square  root  of  4944.9024. 

70.32 

Solution  : 


.49  44.90  24 

49 

140 

0  1 

44  90 

3 

1403 

42  09 

14060 

2  8124 

2 

140 

62 

2  8124 

The  square  roots  are  -f  70.32  and  —  70.32. 

The  first  trial  divisor  is  140.  Since  this  is  greater  than  44,  the  first 
remainder,  annex  0  to  the  root,  obtaining  70. 

The  second  trial  divisor  is  1400;  (2x70=140;  annex  0,  1400). 
Bring  down  the  next  period  90,  getting  for  the  second  remainder  4490. 
Divide  44  by  14  gives  8+  ;  annex  8  to  the  root  and  add  3  to  1400,  etc. 


250  ALGEBRA 

EXERCISE  111 

Find  the  square  roots  of : 

1.  1521.                          5.    23409.  9.  462.25. 

2.  4489.                         6.   54756.  10.  9.8596. 

3.  5625.            ^             7.   173889.  11.  11.9716. 

4.  8836.                         8.    42025.  12.  17.8929. 

181.  The  Approximate  Square  Root  of  a  number  which  is 
not  a  perfect  square  is  often  desired.  Obtain  usually  the  first 
three  figures  following  the  decimal  point. 

Example.     Find  the  approximate  square  root  of  2. 
Solution  :  1.414 


2.00  00  00 

20 
2-t 

1 
100 

96 

28 
21 

0 

4  00 
2  81 

2820 

119  00 

2^ 

ll 

112  96 

7  04 
The  square  roots  are  -}-  1,414+  and  —  1.414+. 

Note.  In  order  to  obtain  the  desired  number  of  decimal  places,  annex 
zeros  until  there  are  three  periods. 

EXERCISE  112 

Find  the  approximate  square  roots  of : 

1.  3.  3.    6.  5.    10.         7.    13.  9.    15.         11.    19. 

2.  5.  4.    7.  6.    11.  8.    14.         10.    17.         12.    21. 

182.  Table  of  Square  Roots.  In  the  remainder  of  the  course, 
it  will  be  necessary  to  use  frequently  the  square  roots  of 
some  numbers.     Retain  some  of  the  square  roots  as  they  are 


SQUARE  ROOT  AND  QUADRATIC  SURDS    251 

found,  either  in  a  notebook  or  in  some  other  convenient  place. 
Make  a  list  of  the  numbers  from  1  to  50,  and  write  their 
square  roots  beside  them,  thus : 

NrMBER  Square  Root 

1  1.000 

2  1.414 

3  1.732 

After  working  Exercise  112,  twelve  of  the  numbers  of  this 
table  may  be  tabulated.  These  roots  may  be  used  to  obtain 
the  square  roots  of  other  numbers. 

Example  1.     Find  the  square  root  of  8. 


Solution  :   \/8=V4x2  =  2x\/2  =  2x  (1.414+)  =  2.828+. 
Example  2.     Find  the  square  root  of  12. 


Solution  :   Vl2  =  V4  x  3  =  2V3  =  2  x  (1.732+)  =  3.464+. 

EXERCISE   113 

1.  Find  the  following  square  roots  to  three  decimals : 

(a)  V18.       (b)  V20.       (c)  V24.       (d)  V27.        (e)  V28. 

2.  Comjilete  your  table  of  square  roots  up  to  50.  Get  as 
many  roots  as  possible  by  inspection  (§  175) ;  get  as  many 
of  the  remaining  roots  as  possible  as  in  Example  1.  Find  the 
others  by  the  long  division  method  (§  180). 

183.  The  square  root  of  a  fraction  which  is  not  a  perfect 
square  may  be  found  as  follows : 


>'2      >'2x2      ^4 


v5^      VO 
V4  2   ' 


±^  =  ±2_i49+^         224+ 
2  2 

Rule.  — To  find  the  square  root  of  a  fraction  : 

1.    Change  the  fraction  into  an  equivalent  fraction  with  a  perfect 
square  denominator. 


1. 

1- 

4. 

i- 

2. 

5 

5. 

i- 

3. 

A- 

6. 

f 

252  ALGEBRA 

2.  The  square  root  of  the  new  fraction  equals  the  square  root  of  its 
numerator  divided  by  the  square  root  of  its  denominator. 

3.  If  desired,  express  the  result  of  step  2  in  simplest  decimal  form, 
prefixing  the  double  sign,  ±- 

ExAMPLK.     Find  the  approximate  square  root  of  |. 
Solution:    1.    The    smallest  square    number  into   which   8   can  be 
changed  is  16  ;  multiply  both  terms  of  the  fraction  by  2. 

^S      >'2  X  8      ^lO  4  4 

EXERCISE  114 
Find  the  approximate  square  roots  of : 

7.  f  10.  f  13.  ^, 

8.  |.  11.  |.  14.  A. 

9.  f  12.  yV  15.  2V 
QUADRATIC  SURDS 

184.  The  indicated  square  root  of  a  number  which  is  not  a  per- 
fect square  is  called  a  Quadratic  Surd ;  as,  V3,  \/- ,  ^x  \/-  +  1- 

^  o '       ^     y 

185.  Surds  should  be  simplified  as  in  the  following  examples: 

Thus,  a  quadratic  surd  is  in  its  simjjlest  form  when  the 
number  under  the  radical  sign  is  an  integer  which  does  not 
contain  any  perfect  square  factor. 

While  a  quadratic  surd  has  two  values,  one  positive  and  one 
negative,  it  is  agreed  to  consider  only  the  positive  root,  in 
order  to  avoid  ambiguity.  This  root  is  called  the  principal 
root. 

186.  Addition  and  Subtraction  of  Surds. 
Example  1.     Find  the  sum  of  V20  and  V45. 

Solution  :  1.    V20  +  V45  =  V4T5  +  V9T5  =  2  VS  +  SVS  =  6V5. 


SQUARE   ROOT  AND  QUADRATIC   SURDS  253 

This  solution  assumes  that  surds  may  be  added  like  other  numbera. 
The  coefficients  of  Vo  are  2  and  3  ;  the  sum  is  found  by  multiplying  V6 
by  the  sum  of  its  coefficients  (§  34). 

The  advanta<ie  in  adding  surds  in  this  way  is  that  fewer  square  roots 
need  be  obtained.  Thus,  the  sum  of  V20and  V-i(>  is  5Vo  or  5x  (2.230+) 
or  11.180+.  This  same  result  could  be  obtained  by  adding  the  square 
roots  of  20  and  45. 

Example  2.     Simplify  V|  +  VJ. 

Solution:  1.    J2+ Jl  =  J5+^=  3v^+ V2^3v^+2V2^5V2^ 
A/8^\2     A/i6^>/4        4  2  4  4  4 

2      5V2^r)X(1.4U+)^7.070+_^.^., 
4  4  4 

Example  3.     Simplify  |  +  V|". 

2  .  ./1_2   ,  ./3^2      V3_2+  V3 


Solution:  1.   ?+Ji=£_j_    /§ 
3^>'3      3      \j) 


3      3      >9     3       3  3 

2     2-1-  V3  ^2  +  1.732+^  8.732+ ^.^^^^^ 
3  3  3  '         * 

Note.  The  results  of  problems  involving  surds  are  often  left  in  the 
surd  form  as  in  step  1  of  Examples  2  and  3.  There  are  advantages  in 
finding  the  approximate  decimal  value  of  the  result. 

EXERCISE  115 
Simplify  the  following : 

1.  V12+V75.  11.  |  +  V|. 

2.  V98-V18.  12.  i-V|. 

3.  VSi")  -  V20.  13.  I  +  V^. 

4.  3V27-V48.  14.  -j  +  VA- 

5.  V28+V63.  15.  _3_4.V^. 

6.  6V2-V50  +  VI8.  16.  +|+V|. 

7.  Vl2+Vi.  17.  -i-\-^l 
8-  Vf+V^.  18.  -f  +  Vf 
«•   Vl  +  V^.  19.  -f  +  VI^ 

10.   V|  -  V24.  20.       f  -  V  V-. 


XV.   QUADRATIC   EQUATIONS 

187.  A  Quadratic  Equation  is  an  equation  of  the  second 
degree  (§  161) ;  it  may  have  one  or  more  unknowns. 

188.  A  Pure  Quadratic  Equation  is  a  quadratic  equation. hav- 
ing only  one  unknown,  which  contains  only  the  second  power 
of  the  unknown,  as,  ax^  =  b. 

Example  1.  An  acre  of  ground  contains  43,560  square  feet. 
How  long  must  the  side  of  a  square  field  be  in  order  that  the 
area  of  the  field  shall  be  one  acre  ? 

Solution  :  1.   Let  s  =  the  number  of  feet  in  one  side. 

2.  Then  s^  =  the  number  of  square  feet  in  the  area. 

3.  Then  s2  ^  43,660. 

Extract  the  square  root  of  both  members  of  the  eciuation. 

4.  Then  s=  ±208.7+. 

Since  this  is  a  field,  only  the  positive  root  has  meaning ;  hence  the 
side  of  the  field  must  be  208.7+  feet. 

189.  A  pure  quadratic  equation  has  two  roots,  because  two 
square  roots  are  obtained  in  extracting  the  square  roots  of  the 
two  members  of  the  equation. 

Rule.  —  To  solve  a  pure  quadratic  equation. 

1.  Clear  the  equation  of  fractions,  transpose,  and  combine  terms 
until  the  equation  takes  the  form  x'^  =  a  number. 

2.  Extract  the  square  root  of  both  members  of  the  equation,  plac- 
ing the  double  sign,  ± ,  before  the  root  in  the  right  member. 

Note.  After  extracting  the  square  roots  of  both  members  of  an  equa- 
tion like  x^  =  a^,  we  get  ±x  =  ±a.  This  gives :  -{-  x  =  -\-  a,  -{-  x=  —  a, 
—  x=+a,  and  —  x=—  a. 

254 


QUADRATIC   EQUATIONS  265 

If  both  members  of  the  last  two  equations  are  multiplied  by  —  1,  the 
equations  become  +  x  =  —  a,  and  -{-  x  =  -\-  a.  These  are  the  first  two 
of  our  four  equations.  Thus,  it  is  clear  that,  from  x'^  =  a'^,  we  get  only 
two  equations,  x  =  -{-  a  and  x  =  —  a^  or  x  =  ±a. 

Example.     Solve  the  equation  — — -  -\ —  =  t^t  H • 

^  3       m      12      m 

Solution:  1.  f^  +  ^  =  ^+^- 

3       w      12      m 

2.  Mxim-.  8m2+36  =  m2  +  144. 

3.  Simplifying  :  7  m^  =  108. 

4.  2>7 :  m^  =  1^^. 

5.  V":*  m=±\/^^=±Qy/}  =  ±^\^. 

6.  a/2I=4.582:  m  =± -•  (4.582)  =  ±  ^Ii^  =  ±  3.927. 

7  7 

7.  mi  =  +  3.1)27;  m2  =-3.927. 

"  wii  "  is  read  "  m  one."  The  numeral  1  is  called  in  such  cases  a  sub- 
script. *'  m2  "  is  read  "  m  two."  These  subscripts  are  used  in  this  case 
to  distinguish  between  the  two  roots  of  the  quadratic. 

Check  :  In  cases  such  as  this,  it  is  better  to  check  by  going  over  the 
solution  a  second  time.  Great  care  must  be  taken,  however,  for  it  is 
easy  to  overlook  an  error. 

Note.  —  Get  the  result  in  the  radical  form  first;  that  is,  m  =  ±  ?  V21  ; 
then  it  is  wise,  for  many  reasons,  to  get  it  in  decimal  form  as  finally  given. 


EXERCISE   116 
Solve  the  following  equations : 

1.  7 p^- 175  =  0.  5.  3(m-2)+2m(m-l)=m. 

2.  5ar^-48  =  80-3a:2^  6.  ^t -{- 5)  -  t(t  -  1)  =  4.  t. 

3.  12c2-140  =  9c2-32.  7.  9a'-7  =  0. 

^    2m^_3jn^-7^11.         g.  lla^-3  =  l. 
5  3  15 

*  The  symbol  "  V    :  "  placed  in  the  left  margin  will  mean,  "  take  the  square 
root  of  both  members  of  the  previous  equation." 


256  ALGEBRA 

Review  the  statement  made  at  the  top  of  page  187;  bear  it 
in  mind  whenever  solving  fractional  equations. 

9    'j_ 11^5  3  ^^+2     5^^-3^4^^-fl 

3/S2     ^S^     6*  "        5  10  25    * 

13  r/+4     gr-4^10 

•  ^_4     ^+4       3 

r*_3r2+4        7^-3 
o.    14.  — 


10. 

n. 

2r- 
4- 

-3       9-f-r 
r       3r4-2 
x-\-l      x"^— 

x  +  1 

x  +  1  3r*-i-2?-2-4     3?'2  +  2 


15.   a2-2ca^  = 

=  3  Ir.     Solve  for  ic. 

Solution:  1. 

-  2  0x2  zz  3  62  _  a2. 

2. 

2  ca;2  =  a2  _  3  ^,2. 

3. 

^2_«2-3&2 

2c 


=  ±  —  V2  a2c  _  6  62c. 
2c 
Solve  for  x : 

P,  jl6.    a2+a;2  =  m.  19.   2cx^=cP, 

17.  a4-2a;^  =  c.  20.   i^ar^  =  c. 

18.  aa^  =  m.  21.   3m3^  —  n=p, 

190.   A  Right  Triangle  is  a  triangle  which  has  a  right  angle 
for  one  of  its  angles ;  as  triangle  ABC,  in  which  angle  jB  is  a 
'  [  right  angle. 

The  side  opposite  the  right  angle  is  the  hypote- 
^b  nuse;  as,  side  AC.     The  side  BC  is  the  base  and 

AB  is  the  altitude. 

In  a  right  triangle,  the  square  of  the  hypotenuse 
equals  the  sum  of  the  squares  of  the  other  two  sides. 
Thus,  b'  =  a'-\-  cl 

To  verify  this  fact,  draw  a  right  triangle  with  BC  3  inches 
m^  AB  4  inches;  measure  AC.  Substitute  the  lengths  of  the 
sides  in  the  equation  b^  =  a^  +  cl 


QUADRATIC   EQUATIONS  257 

EXERCISE  117 
Carry  out  all  results  in  this  set  to  one  decimal  place: 

1.  Find  the  altitude  of  a  right  triangle  whose  base  is  13  feet 
and  whose  hypotenuse  is  30  feet. 

Solution  :  1.   Let  x  =  the  number  of  feet  in  the  altitude.  ' 

2.  Then  x^  4-  132  ^  302,  (^l^y  ?) 

3.  Complete  the  solution. 

2.  Find  the  base  of  a  right  triangle  whose  hypotenuse  is 
45  feet  and  whose  altitude  is  25  feet. 

3.  If  the  diagonal  of  a  rectangle  is  68  inches  and  the  base 
of  the  rectangle  is  three  times  the  altitude,  what  are  the  di- 
mensions of  the  rectangle  ? 

4.  If  the  altitude  of  a  rectangle  is  a  and  the  base  is  4  timias 
the  altitude,  what  is  the  length  of  the  diagonal? 

5.  Solve  the  formula  W  =  d?  +  (? :  (a)  for  a  ;  (6)  for  c. 

In  the  isosceles  triangle  ABC,  AD,  which  is  perpendicular  to 
BC,  is  the  altitude  and  BC  is  the  base.  BD  and  i?C  are  equal; 
this  may  be  verified  by  measuring  them.  ^ 

6.  If  AB  is  15  inches  and  BC  is  18  inches, 
find  ^W. 

7.  If  the  equal   sides  of  an  isosceles  tri- 
angle are  each  30  inches,  and  the  altitude  is  18  inches,  find 
the  base. 

8.  If  the  equal  sides  of  an  isosceles  triangle  are  each  3  a 
inches  and  the  base  is  2  a  inches,  find  the  altitude  ? 

9.  An  equilateral  triangle  is  one  which  has  all  of  its  sides 
equal.  Find  the  altitude  of  an  equilateral  triangle  whosQ  sides 
are  each  10  inches. 

10.  Find  the  altitude  of  an  equilateral  triangle  whose  sides 
are  each  s  inches;  (b)  find  the  area  of  the  triangle. 

11.  The  area  of  a  circle  is  found  by  the  formula  A=7r7'^, 
where  r  is  the  radius,  and  tt  =  3|.  f 

Find  the  area  of  a  circle  whose  radius  is  9  inches.. 


258  ALGEBRA 

12.  What  is  the  radius  of  a  circle  whose  area  is  an  acre  ? 
(Express  the  radius  in  feet.     See  §  188.) 

Express  the  results  of  the  following  in  simplest  radical  form). 

13.  Solve  the  formula  A  =  ttt^  for  r:  (a)  letting  7r=3|; 
(6)  without  substituting  the  value  of  tt. 

14.  The  volume  of  a  circular  cylinder  is  given  by  the  for- 
mula F=  TTT^h  ;  where  r  =  the  radius,  h  the  altitude. 

Find  V  when  r  =  5  and  h  =  13. 

15.  Find  r  when  V=  759  and  h  =  14. 

16.  Solve  the  cylinder  formula  for  r. 

17.  Solve  the  formula  S  =  4:  irr^  for  r. 

18.  Solve  the  formula  V  =  ^  irr^h  for  r. 

19.  The  distance  s  in  feet  through  which  an  object  will  fall 
in  t  seconds  is  given  by  the  formula  s  =  i  gt^,  where  g  is  32. 

Suppose  that  a  stone  is  allowed  to  fall  from  a  tower;  how 
far  will  it  fall  in :   (a)  3  seconds  ?     (6)  5  seconds  ? 

20.  How  long  will  it  take  a  ball  to  fall  900  feet? 

21.  Washington's  Monument  in  Washington,  D.  C.  is  555  feet 
high.     How  long  will  it  take  a  ball  to  fall  that  distance  ? 

22.  Solve  the  equation  s  =  \gt^  for  t. 

23.  Solve  the  formula/=^^  for  V. 

COMPLETE  QUADRATIC   EQUATIONS 

191.  A  Complete  Quadratic  Equation  is  a  quadratic  equation 
having  only  one  unknown,  which  contains  the  first  power  of 
the  unknown  as  well  as  the  second  power;  as, 

23^-3x-5=0. 

192.  Complete  quadratic  equations  have  been  Solved  by  Fac- 
toring in  §  108. 


QUADRATIC   EQUATIONS  259 

EXERCISE   118 
Solve  by  factoring : 

1.  ar-5a;  =  24.  ;^3    J__Jt5_^_l 

o02  if:a  '8ar^24x         2* 

2.  2  ??r  —  m  —  15  =  0. 

3.  2a:2  +  i3^._^20  =  0.  14.   ^4-1  =  -??. 

4.  12/- 77/  =  10.  ^     "^^  ^^ 

5.  5  lu^  —  7  w  =  0. 

6.  9a,'2-49x  =  0. 


(Solve  for  j-). 


15. 

1-t' 

12 
3-t 

16. 

7 

s-2 

s-3 

s-4 

17. 

7 

3 

1 


8.  8  ic^  +  14  7/ia:  =  —  3  m^  '  y  __  3     2^  —  4      2 

9.  6  ar»- 11  a!A;  =  10  Ar'.  ,  2  6 

18.  1+—=^ ^^=0. 

10.   5ar^  +  152>'  =  28icp.  P-3      i>  -  8 

a^     a;     35  lo        3  2     _5 


19. 


326  7^(1, +  6m  —  54 

12.  1_A=_^.  20.  ?:ii?=^^ 1. 

2      ^x"         12  a;  r  +  3      (r-hS)^ 

193.  Graphical  Solution  of  Equations  With  One  Variable.  Many 
facts  about  equations  containing  one  variable  can  be  discovered 
by  the  aid  of  graphical  rejjresentation. 

Example  1.    Consider  the  equation  3  a;  —  12  =  0. 
The  expression  Sx  —  12  has  a  different  value  for  each  value  of  x: 
Thusif  x  =  2,  3x-12=-6;  if  x  =- 3,  3x  -  12  =- 21. 
The  problem  is  to  find  the  value  of  x  for  which  the  expression  3  x  —  12 
will  equal  zero. 

Graphical  Solution  :  1.   Let  y  =  3  x  —  12. 

2.    Find  values  of  y  for  some  values  of  x : 

if  X  =  0,  ?/ =- 12.  if  x=-2,  y  =- 18. 

.    if  X  =  +  5,  y  =  +  3.  if  X  =  +  6,  y  =  +  6. 


260 


ALGEBRA 


3.  Use  these  pairs  of  numbers  as  coordinates  of  points  and  draw  the 
graph. 


■     1    1    1     V      '              M    I/C 

1                                      1    \l\ 

1                                        / 

J-                J-^m-                   -     Z     - 

j  ■+!<)           ~  r 

1              / 

1              f 

-'ib          r^    - 

1           ( 

1          j* 

1              1        *  /            1    .- 

X    ,.             1          1    1     0        A/          _^__^            j_  _X 

-   )           II-                  1                1  ji   5                  13 

1    i                    i                1/ 

1    i                    1               / 

t: 

__  _          ^     Z                    : 

^ 

^    -U 

.x 

1    in    /                                                   1 

1 

n) 

J   , 

Sl  y'" 

Fig.  9 


4.   jBC  crosses  the  x  axis  at  point  A.     The  coordinates  of  ^  are :  x  =  4, 

2/  =  0. 

6.    Hence  when  x  =  4,  3  x  —  12  =  0.     (y  is  the  expression  3  x  —  12.) 
.-.  X  =  4  is  the  desired  solution  of  the  equation,  for  we  were  looking  for 

a  value  of  x  for  which  3  x  —  12  =  0. 

Rule.  —  To  solve  graphically  an  equation  containing  one  variable : 

1.  Simplify  the  equation  as  much  as  possible. 

2.  Transpose  all  terms  to  the  left  member. 

3.  Represent  by  y  the  expression  found  in  step  2. 

4.  Find  for  y  the  values  which  correspond  to  selected  values  of 
the  variable  in  the  equation. 

6.  Use  the  pairs  of  values  obtained  in  step  4  as  coordinates  of 
points ;  plot  the  points ;  draw  the  graph,  making  the  vertical  axis 
the  y  axis. 

6.  The  graph  crosses  the  horizontal  axis  at  points  whose  ordinates 
are  zero,  and  whose  abscissas  are  the  desired  roots  of  the  equation. 


QUADRATIC   EQUATIONS 

Example  2.     Solve  the  equation  ar  —  a;  =  6. 

Solution  :  1.   x^  —  ar  =  6,  or  x"^  —  x  —  0  =  0. 

2.  Let  y  =  X-  —  X  —  a. 

3.  If  a;=-4,  y=  (-4)2-  (_  4) -6=  10  +  4-6  =  4-14. 
4. 


261 


Similarly  \f  x  = 

0 

+  1 

+  2 

+  4 

+  5 

-  1 

-2J-3 

-4 

then  y  = 

-6 

-6 

-4 

+  6 

+  14 

-4 

0 

+  6 

+  14 

■MllUllllIM 

iliiii 

::::  "  "  ~  5     :        -^  -     --     -> 

+  — S / : 

^— >^ 

_J ^ ^4^^  J LJ__^ 

1 L-te 1 

::    :::     :::  ::x'-::::::  ::    .: 

Fig.  10 


5.  The  graph  crof3ses  the  horizontal  axis  at  the  points  ^jand  B.  Ac- 
cording to  the  rule,  the  abscissaB  of  these  points  are  the  two  roots  of  the 
equation. 

At^:  X  =  -  2,  y  =  0  ;  i.e.  x^  -  x  -  (i  =  0.  I 

AtB:  x=-\-S,y  =  0;  i.e.  x^-x-(i  =  0. 

Check  :        x  =  -  2  ;  does  (_  2)2  _  (_  2)  -  6  =  0  ?     Yes. 
x  =  +  3;  does  (+3)2- (+3) -6  =  0?:  Ted. 


262 


ALGEBRA 


EXERCISE  119 

9olve  graphically  the  equations : 

1.  ic  H-  3  =  0.  4.   a^  -  a;  =  12. 

2.  2 ic  =  7.  5.    x^—7x-\-6  =  0. 

3.  x^  =  16.  6.   x'-\-6x-\-5  =  0. 

194-  Some  quadratic  equations  cannot  be  solved  readily  by 
factoring ;  for  example,  ic^— 6cc  —  2  =  0,  since  x^  —  6x  —  2 
does  not  have  any  rational  factors.  The  graphical  solution 
shows  that  this  equation  has  two  roots. 

Solution:  1.    ic2_6a;-2  =  0. 


2.    Lety  =  a;2- 

-6x- 

-2. 

When  X  = 

-2 

-1 

0 

+  1 

+  2 

+  3 

+  4 

+  6 

+  7 

+  8 
+  14 

then  y  = 

+  14 

+-5 

-2 

-  7 

^10 

-11 

-10 

-2 

+  5 

y:                                                   1 

'U-                                              Jl 

I                                                  4 

x                        t 

i  *«^n:                   -  L 

jL:E     it                   t: 

it                                       41 

V                                              J 

\                               I 

j  ,,         _                  r  _ 

\                             T 

t                                             4 

A                                       A 

.7                                                           ^         ^               I-                                               JL- 

X A  ^    '                           4^         ^ 

-i    -."^     -;     -      \     i+l   i+2   +3  +'    +     +')/+7    "t 

:     _L_              I 

it    L-                                                                 X 

i               i 

i^              ^'^ 

^^  r                7 

>^               ^ 

!^                7 

^^           V 

-^^         S        ^Z^- 

-',°         ^  -  --k 

1  i  . , 

J- XL    -- 

Fig.  11 


3.   The  graph  crosses  the  horizontal  axis  at  the  points  A  and  B.    The 
abscissa  of  A  is  about  —  .3  ;  the  abscissa  of  JB  is  about  +6.3. 


QUADRATIC   EQUATIONS  263 

This  indicates  that  the  roots  of  this  equation  are  approximately  —  .3, 
and  +  6.3. 

There  are  two  methods  of  solving  such  equations  which  give 
the  roots  more  accurately. 

195.   Solution  by  Completing  the  Square. 
Development  1.     Find:  (a)  (x  — 4)^;  (b)  (a? +  5)^; 

(<-■)  (^-r;  (<*)  (y+if- 

2.  When  is  a  trinomial  a  perfect  square  ?     (See  §  96.) 

3.  Make  a  perfect  square  trinomial  of  ic^  —  10  a;. 

Solution  :  1.    ^  of  10  =  5  ;  62  =  26  ;  add  25. 

2.    The  perfect  square  is  x'^  —  10  ic  -f-  26  or  {x—  5)2. 

4.  Make  perfect  square  trinomials  of  the  following : 

(a)  ar'-12a?;  (b)  y^-Uy,  (c)  z^-20z. 

5.  Solve  the  equation  a^  —  12  a;  +  20  =  0. 
Solution  :  1.     S20  '■  a;2  -  12  a;  =  -  20. 

2.  Make  the  left  member  a  perfect  square  by  adding  36 ;  therefore 
add  36  to  both  members  :  (§41). 

^36  :  x2  -  12  X  +  36  =  36  -  20. 

or  (x  -  6)2  =  16. 

3.  V":  x-e=±4. 

4.  .'.  a;  —  6  =  +  4,  or  a;  =  6  +  4  =  10,  one  root, 
and  x  —  6=—  4,  orx  =  6  —  4  =  2,  another  root. 

Check  :       x  =  10  ;  does  (10)2  -  12(10)  +  20  =  0  ?    Yes. 
x  =  2]    does  (2)2  -  12(2)  -f  20  =  0  ?    Yes. 

6.  Solve  the  equation  a^  —  Sx  —  5=  0. 
Solution  :   1.  x2  —  3  x  —  6  =  0. 

2.  As:  x^-Sx=+b. 

3.  i(-  3)  =  -  I ;  (-  f )«  =  +  f  ;  add  I  to  both  members. 

4.  A^:         x2-3x  + J  =  6  +  |  =  ^. 

6.    V:  x-^  =  ±V'^-  =  ±iy/29. 

6.  ,.^  =  3      1^29=3-±^. 

2     2  2 


264  ALGEBRA  ; 

7.  Radical  results,      x-^  =  ^jty^  and  X2  =^-^^. 

2ii  2 

8.  Decimal  results,     xi  =  ^  +  ^-^^^  and  x^  =  ^~f-^^^ . 

_  8.885  ^  -  2.385 

2  ~        2 

=  4.192+  =-1.192+. 

Check  :  To  check  the  solution  by  substituting  the  roots  in  either  their 
decimal  or  their  radical  form  is  a  long  process,  with  many  opportunities 
for  errors.  Persons  skillful  in  algebra  check  by  going  over  the  solution 
carefully. 

A  quick  check,  the  reason  for  which  will  be  learned  later  in  algebra,  is 
to  find  the  algebraic  sum  of  the  roots  ;  this  result  should  equal  the  nega- 
tive of  the  algebraic  coefficient  of  x  in  the  equation  in  which  the  coefficient 
of  x^  is  1. 

Here:  +4.192+  The  coefficient  of  cc^  is  1.    The  coefficient  of  x 

—  1.192+       is  —  3.     This  equals  the  negative  of  the  algebraic 

Sum.     +  3  sum  of  the  roots. 

If  the  coefficient  of  x^  is  not  1,  first  imagine  the  equation  divided  by 
that  coefficient,  and  then  select  the  coefficient  of  x. 

Rule.  —  To  solve  a  quadratic  equation  by  completing  the  square : 

1.  Simplify  the  equation;  transpose  all  terms  containing  the 
unknown  number  to  the  left  member,  and  all  other  terms  to 
the  right  member  so  that  the  equation  takes  the  form 

ax^  -\-bx=c. 

2.  If  the  coefficient  of  3^  is  not  l,  divide  both  members  of  the 
equation  by  it,  so  that  the  equation  takes  the  form 

Z,  Find  one  half  of  the  coefficient  of  x\  square  the  result; 
add  the  square  to  both  members  of  the  equation  obtained  in  step  2. 
This  makes  the  left  member  a  perfect  square. 

4.  Write  the  left  member  as  the  square  of  a  binomial ;  express 
the  right  member  in  its  simplest  form. 

5.  Take  the  square  root  of  both  members,  writing  the  double 
sign,  ± ,  before  the  square  root  in  the  right  member. 


QUADRATIC   EQUATIONS  265 

6.  Set  the  left  square  root  equal  to  the  +  root  in  the  right 
member  of  the  equation  in  step  6.  Solve  for  the  unknown.  This 
gives  one  root. 

7.  Repeat  the  process,  using  the  —  root  in  step  5.  This  gives 
the  second  root  of  the  equation. 

8.  Express  the  roots  first  in  simplest  radical  form,  and  then,  if 
desired,  in  simplest  decimal  form. 

EXERCISE  120 
Solve  by  completing  the  square : 

10  z^-10z  =  5. 

11.  x^-\-5x=  —^. 

12.  s2_|.io  =  7s. 

13.  w;2  -  3  w  ==  -  2. 

14.  m2-f-m  =  30. 

15.  7-2-13r-|-30=0. 

16.  z^-6  =  5z. 

17.  ^2  +  9^  =  11. 

9.  t^-St=4:.  18.   /-152/-fl6  =  0. 

19.   Solve  the  equation  7?  —  ^x  =  l, 
Solution  :  1.  i  of  f  =  \.        {\Y  =  \. 

2.  Aj:  a;2-|x-f  i=l  +  i. 

3.  {.x-\y  =  {^)-_ 

5.          x-\=^-\V^.  x-\=-\\/\0. 

x  =  i  +  ^ViO  x  =  i-i\/lO 

_1  +  VlO  _  1  -  VlO 

~        3  3 

_  1  +  3.162  ^1-3.162 

O  Q 

3  3 


1. 

ar^_4a;-5  =  0. 

2. 

ar^  +  8  a;  -  33  =  0. 

3. 

ic2_6a;-27  =  0. 

4. 

x2  +  10ic4-24  =  0. 

5. 

ar^_2a;-15  =  0. 

6. 

2/s_42/  =  4. 

7. 

a2  +  6a  =  l. 

8. 

m^  -  2  m  =  1. 

266  ALGEBRA 

Check  :  +  1.387  Coefficient  of  a:  =  —  |  =  —  .6666. 

-    .720  =  _  .667. 

Sum.        +    .667  +.667  = -(-.667). 

20.  x^-^x^^.  23.    a^-la  =  ^. 

21.  f-iy  =  ^.  24.   m2-fm  =  |. 

22.  z^-\-^z  =  l.  25.    f-^t  =  T. 
26.    Solve  the  equation  3ic^  —  2aj  —  1=0. 


Solution 

:   1.    Da: 

2. 

H-' 

3. 

Complete  the  solution  £ 

27. 

3a^ 

-  2  a;  -  5  =  0. 

28. 

or'-\-2r-S  =  0. 

29. 

4.f- 

-8^  +  3  =  0. 

30. 

Sx'- 

-4aj-7  =  0. 

31. 

2  a'. 

-  3  a  -  9  =  0. 

32. 

5p'-^Sp  =  l. 

33. 

6w' 

-5w  =  10. 

34. 

3 

5  a' 

-2-5=0 
a 

38. 


t         St     5 


^-3      4      2 


39.    -^  +  1  =  0. 
«— 4     5 


42. 


Sw 

w  —  1 

+ 

2 

4 

1 

a  — 

5 

— 

2 
a  — 

2^ 

2 

1 

40.  -i^i^+_-:^=0. 
41. 


3 
5* 

15 

c-2      c+2  8* 


1  V.O  3                 2            . 

35.^+2-2^  =  0.  43.  _^___  =  1. 

36.  !^_2+4  =  0.  44.  ^-^  +  A=     2     . 
3      m  22/-7     15     2/-3 

37.  I  +  ?-5  =  0.  45.  2            ^-            1 


r'r  to^  —  9w-f-3w  —  3 

196.    Solution  of  Literal  Quadratic  Equations. 
Example.     Solve  the  equation  ax^  —  3  6a;  —  c  =  0. 
Solution  :  1.  aoH^  —  3  &a:  —  c  =  0. 


QUADRATIC   EQUATIONS  267 


2.  D,:  x^-  —  x^^  =  0. 

a         a 

3.  A,:  a;2_§^x  =  ^. 

-        .  a         a 

4.  The  coefficient  of  x  is  (^-^)  ;  one  half  of  it  is  (^r^)- 
The  square  of  (^^^-^)  is  l^^\  •    Add  this  to  both  members  of 


equa- 


tion 3. 


Ix      3&\2^96^  +  4, 
^        2a)  4a2 


36 


6.  a;-~  =  ±r^V962  +  4ac, 

2  a        2  a 


7  ^  _  +36  j:  V9  62  +  4  ac 

I,  X  —  

2a 

Q      .    ^       +  8  6  +  \/9  62  +  4  ac  .   ^  _  +  3  6  -  V9  62  +  4  ac 
2a  2a 

Check  :  xi  +  iC2  =  -^^^ — -  =  H Since  this  is  the  negative  of  tho 

2a  a 

coefficient  of  x  in  step  2,  the  roots  are  correct. 

EXERCISE   121 
Solve  the  following  equations  for  x : 

1.  x^  +  2ax-S  =  0.  8.   x^-\-2'mx  =  2m  +  l. 

2.  x^-\-2ax-\-b  =  0.  9.   x"- 4.ax  =  9b^-4:a\ 

3.  x^-{-4tX-c  =  0.  10.    aa^4-2a;  +  l  =  0. 

4.  xr-i-Sx+7rt  =  0,  11.    ra;^  +  4 ^a;  —  5  =  0. 

5.  2a;=^4-3a;-n=0.  12.    car^  +  2 c?a;  +  p  =  0. 

6.  2x^-^4:  ax -\-b  =  0.  13.   .-c^  +  ma;  +  n  =  0. 

7.  x^  +  3ax-4:t  =  0.  14.    aa;^  _f_  ^^  ^  ^  ^  0 

197.    Solution    of  Quadratic    Equations   by   a   Formula.      All 

quadratic  equations  having  one  unknown  may  be  put  in  the 
form  ax'  +  bx  +  c^O. 


268  ALGEBRA 

This  equation  may  be  solved  like  the  equations  in  Exercise 
121.     The  roots  will  be  found  to  be 


^  2a 

This  result  may  be  used  as  di  formula  for  solving  any  quad- 
ratic equation  of  the  form  ax^  -\-hx-\-c. 

Example  1.     Solve  the  equation  2a^  —  3aj—  5  =  0. 
Solution  :  1.    Comparing  the  equation  with  ax^  +  6a;  +  c  =  0 : 

a  =2,    6  =-3,  c=-5. 
2.   Substitute  these  values  in  the  formula : 


-h±  V&2  -  4  ac 
2a 


3.  Then  x  =  -(-3)±  V(- 3)^-4(2)(- 5) 

_2(2) 

_  +  3  zt:  V9  +  40 

4 
_  3  ±  \/49  ^  3  j:  7 
4  4 

4.  ..  xi-     ^     -^_2,a:2-     ^     -    ^    -      1. 

Check  :  a^i  =  f  ;  does  2 (|)2  _  3(f) -  6  =  0 ? 
does2.^:f  -  V---6  =  0? 
does  Y-V- 5  =  0?    Yes. 
X2=-l;  does  2(- 1)2 -3(-l)- 5  =  0? 
does  2  +  3-5  =  0?    Yes. 

Example  2.     Solve  the  equation  2ic2_3a._3_.0- 

Solution:!.  a  =  2,    6  =—3,    c=— 3. 

2.   Substituting  in  the  formula,  x  —  — — ^— - — — — ~: 

za 

_3j-v/9^-24_3rb  V33  _  3  +  5.744+ 
^~  4  ~        4        ~  4  * 

8.744+      0  1QA+     ^        -  2.744+  ^q^, 

.'.  xi  = =  2.186+  ;  X2  = =—  .o8d+. 


QUADRATIC   EQUATIONS  269 

Check:     2.186+  The  coefficient  of  x  is  —  |,  when  the  coeffl- 

-  .686+  cientof  a;2  =  1  ;  1.5  =_(_  |)_ 

1.600  (For  this  method  of  checking,  see  §  195.) 

EXERCISE  122 
Solve  the  equations : 

1.  4r2-7r  +  3  =  0.  12.   3id^ -T w-^2  =  0. 

2.  6<--f  13^  +  5  =  0.  13.   5r  +  8c  =  4. 

3.  2/2  =  62/4-72.  14.   9x'  +  16x  +  S  =  0. 

4.  a2_7a-30  =  0.  15.   15^2-22^-5  =  0. 

5.  3ar^-2a;-33  =  0. 

6.  3m2-f-5m-f  1  =  0. 

7.  os^  =  5s  —  l. 

8.  6w'-llw-\-2  =  0. 

9.  23^-3x-l  =  0. 

10.  a^  +  a;-l  =  0. 

11.  2/2-4^  +  2  =  0. 

198.    Summary   of   Methods   of   Solving   a   Quadratic.      Four 

methods  of  solving  a  quadratic  equation  have  been  given : 
the  graphical,  by  factoring,  by  completing  the  square,  and  by 
the  formula.  The  first  is  useful  mainly  as  a  means  of  illus- 
tration ;  the  third  is  useful  mainly  in  solving  the  general  quad- 
ratic ax^  +  bx-\-c  =  0,  and,  thus,  in  deriving  the  formula. 

Historical  Note.  Greek  mathematicians  as  early  as  Euclid  were  able 
to  solve  certain  quadratics  by  a  geometric  method,  about  which  the  student 
may  learn  when  he  studies  plane  geometry.  Heron  of  Alexandria,  about 
110  B.C.,  proposed  a  problem  which  leads  to  a  quadratic.  His  solution  is 
not  given,  but  his  result  would  indicate  that  he  probably  solved  the  equa- 
tion by  a  rule  which  might  be  obtained  from  the  quadratic  by  completing 
its  square  in  a  certain  manner.  Diophantus,  275  A.r>. ,  gave  many  problems 
which  lead  to  quadratic  equations.    The  rules  by  which  he  solved  his 


16. 

3      2     2 

17. 

a;  —  -  =  0. 

X            3 

18. 

,^         2^  =  0. 

19. 

2  ^  =  5. 
d-1     d 

270  algp:bra 

equations  appear  to  have  been  derived  by  completing  the  square.  He 
considered  three  separate  kinds  of  quadratics.  He  gave  only  one  root  for 
a  quadratic,  even  vi^hen  the  equation  had  two  roots. 

The  Hindu  mathematicians,  knowing  about  negative  numbers,  con- 
sidered one  general  quadratic.  Cridharra  gave  a  rule  much  like  our 
formula.  The  Hindus  knew  that  a  quadratic  has  two  roots,  but  they  usu- 
ally rejected  any  negative  roots. 

The  Arabians  went  back  to  the  practice  of  Diophantus  in  considering 
three  or  more  kinds  of  quadratics.  Mohammed  Ben  Musa,  820  a.d,,  had 
five  kinds.  He  admitted  two  roots  when  both  were  positive.  Alkarchl 
gave  a  purely  algebraic  solution  of  a  quadratic  by  completing  the  square, 
and  refers  to  this  method  as  being  a  diophantic  method. 

In  Europe,  mathematicians  followed  the  practice  of  the  Arabians,  i^nd  by 
the  time  of  Widmann,  1489,  had  twenty  four  special  forms  of  equations. 
These  were  solved  by  rules  which  were  learned  and  used  in  a  mechanical 
manner.  Stifel,  1486-1567,  finally  brought  the  study  of  quadratics  back 
to  the  point  that  had  been  reached  by  the  Hindus  one  thousand  years 
before.  He  gave  only  three  normal  forms  for  the  quadratic  ;  he  allowed 
double  roots  when  they  were  both  positive.  Stevin,  1548-1620,  went  still 
farther.  He  gave  only  one  normal  form  for  the  general  quadratic,  as  do 
we  ;  he  solved  this  in  both  a  geometric  and  an  algebraic  manner,  giving 
the  method  of  completing  the  square.     He  allowed  negative  roots. 

EXERCISE   123 

Miscellaneous  Examples 

Solve  the  following  equations  by  any  of  the  preceding 
methods.  As  a  rule,  solve  by  factoring  if  possible ;  otherwise 
by  the  formula. 

1     (3  a;  +  2)  (2  a?  -f-  3)  =  (2;  --  3)  (2  a;  -  4). 

2.    9(?/-l)2-4(2/-2)2  =  44. 

^    30         30     _^  g    ic-f-2     4.-x_l 


m      m  -f-1  x—1        2  X       3 

_3 ^==1,  7.  -1^L_ 2__^ 

a  — 6      a  — 5       '  2a-f-5      a  —  1 

_5 ^  =  6  8    2-3^      4-^^11 

x-l-4     x-2       '  *        4          t-2      4 


QUADRATIC  EQUATIONS                          271 

3a  — 5     2a-f  5  _-.  --     2  w -\- S     ^  —  ^_2 

2a  — 5     3a-f5  5           w;  — 4 

4v-3         2v  7-r       2r  +  l 

14  2 


20. 


13. 


x-e     3(a;-l)         3x 


14    y  +  ^  I  y-2^6y  +  16_ 
'   2^-2      ?/  +  2  3i^ 

15.  ^-+J— ^  =  -1 
1-t'^l  +  t     1-t         8 

3r-6      7       11 -2  r 


16. 


5-r       2     2(5  - 2  r) 


3-2a;     2  +  3x^1      IGx  +  a^ 
2  +  a;        2-a;  ~3        aj2-4 

18.  .V  +  1      y  +  2^2y  +  13 
7J-1     y-2        y-1 

19.  l-f---I^_+ ?J^ =  0. 

3a;  +  l^(3a;  +  l)(7a;+l) 

1  15a; 


(3a!+l)(l-5a;)'    2(1-5  a;)(7a;+l)      (3a;+l)(7a;+l) 


21.    Solve  the  equation  2  p^x^  —  3  pqx  —  q^  =  0. 
Solution  .  1.   a  =  2p^  ;  h  =  (-  Spq)  ;  c  =  (-  q"^). 


2.   Formula:  ^  ^-b  ±Vb^  -  i  ac  ^ 

2a 

^  -  (-  3pg)  zb  V(-  Spq)^  -  4(2p2)(-  g2)  ^  +  3pq±  V9pV  +  8pV 
2(2  p2)  4p'^ 


f_Spq  ±  Vn p'^q^  ^  Spq±pqy/17  ^  pq(S  ±  VlT)  ^  g(3  j,  VlT) 
4|)2  41)8  41)8  4i) 


272  algp:bra 

Solve  the  following  equations  for  x : 

22.  x^-6cx-{-5c^  =  0.  27.    2  P-2  Px -{- x^  =  l. 

23.  3x^=:2rx-\-2r^.  28.   x'-(a-l)x  =  a. 

24.  a;2  +  c2  — 2  ajp  =  0.  29.    oi^  +  ax  +  bx -{- ab  =  0. 

25.  x^-xp-\-(c^-d^)  =  0.       30.   ax^-(a-2b)x=z  +  2b. 

26.  lgx^-hax=s.  31.    ma;2+ (2mn-3  n)aj-6  7i2=0. 

EXERCISE   124 
Review  §  112  before  solving  this  set  of  examples. 

1.  Twice  the  square  of  a  certain  number  equals  the  excess 
of  3  over  that  number.     Find  the  number. 

2.  If  three  times  the  square  of  a  certain  number  be 
increased  by  the  number  itself,  the  sum  is  10.  Find  the 
number. 

3.  Find  two  consecutive  integers  whose  product  is  306. 

4.  If  the  product  of  three  consecutive  integers  be  divided 
by  each  of  them  in  turn,  the  sum  of  the  three  quotients  is  74. 
Find  the  integers. 

5.  The  sum  of  the  squares  of  two  consecutive  integers  is 
685.     Find  the  integers. 

6.  The  sum  of  a  certain  number  and  its  reciprocal  is  ^. 
Find  the  number. 

Hint  :  The  reciprocal  of  a  number  is  obtained  by  dividing  1  by  the 
number.    The  reciprocal  of  a;  is  -  • 

X 

7.  Find  the  dimensions  of  a  rectangle  whose  area  is  357 
square  feet  if  its  length  exceeds  its  width  by  4  feet. 

8.  The  numerator  of  a  certain  fraction  exceeds  its  denomi- 
nator by  3.  The  fraction  exceeds  its  reciprocal  by  ff .  Find 
the  fraction. 

9.  The  main  waiting  room  of  the  Union  Railway  Station 
in  Washington,  D.C.,  has  an  area  of  28.600  square  feet.  The 
length  exceeds  the  width  by  90  feet.     Find  the  dimensions. 


QUADRATIC   EQUATIONS  273 

10.  Find  the  base  and  altitude  of  a  triangle  whose  area  is 
63  square  inches,  if  the  base  exceeds  twice  the  altitude  by  4 
inches. 

11.  Find  the  dimensions  of  a  rectangle  whose  area  equals 
that  of  a  square  of  side  24  feet,  if  the  difference  of  the  base 
and  altitude  of  the  rectangle  is  14  feet. 

12.  Find  the  dimensions  of  a  rectangle  if  its  area  equals 
that  of  a  square  of  side  35  feet,  if  the  difference  of  the  base 
and  altitude  is  24  feet. 

13.  Find  the  dimensions  of  a  rectangle  whose  area  is  3750 
square  feet,  if  the  sum  of  its  base  and  altitude  is  155  feet. 

14.  Find  the  dimensions  of  a  rectangle  whose  area  is  1701 
square  feet,  if  the  sum  of  its  base  and  altitude  is  90  feet. 

15.  Find  the  dimensions  of  a  right  triangle  if  its  hypotenuse 
is  20  feet  and  the  base  exceeds  the  altitude  by  4  feet. 

16.  Find  the  dimensions  of  a  right  triangle  if  its  hypotenuse 
is  26  feet  and  the  sum  of  whose  base  and  altitude  is  34  feet. 

17.  Find  the  sides  of  an  isosceles  triangle  if  the  perimeter 
is  35  inches  and  if  the  number  of  inches  in  the  base  is  the 
quotient  of  75  divided  by  the  number  of  inches  in  one  of  the 
sides  of  the  triangle. 

18.  A  man  travelled  105  miles.  If  he  had  gone  9  miles 
more  an  hour,  he  would  have  performed  the  journey  in  1^  hours 
less  time.     Find  his  rate  in  miles  an  hour.     (See  p.  105.) 

19.  If  a  man  travels  120  miles  by  one  train  and  returns  on 
a  train  whose  rate  is  10  miles  an  hour  more,  he  will  require  7 
hours  for  the  trip.     What  is  the  rate  of  the  first  train  ? 

20.  A  crew  can  row  8  miles  downstream  and  back  again  in 
4|  hours ;  if  the  rate  of  the  stream  is  4  miles  an  hour,  find  the 
rate  of  the  crew  in  still  water.     (See  §  144.) 

21.  A  man  travels  10  miles  by  train.  He  returns  by  a  train 
which  runs  10  miles  an  hour  faster  than  the  first,  accomplish- 


274  ALGEBRA 

ing  the  whole  journey  in  50  minutes.     Find  the  rate  of  the 
first  train. 

22.  A  tank  can  be  filled  by  two  pipes  running  together  in 
2  hours.  The  larger  pipe  by  itself  will  fill  it  in  3  hours  less 
time  than  the  smaller  pipe.  How  long  will  it  take  each  pipe 
to  fill  the  tank  alone  ?     (See  §  142.) 

23.  Some  boys  are  canoeing  on  a  river,  in  part  of  which  the 
current  is  5  miles  an  hour,  and  in  another  part  3  miles  an  hour. 
If,  when  going  downstream,  they  go  4  miles  where  the  current 
is  rapid  and  8  miles  where  it  is  less  rapid  in  a  total  time  of 
1|  hours,  what  is  their  rate  of  rowing  in  still  water  ? 

24.  I  have  a  lawn  which  is  60  by  80  feet.  How  wide  a 
strip  must  I  cut  around  it  when  mowing  the  grass  to  have  cut 

half  of  it  ? 

Hint  :  Referring  to  the  figure,  it  is  clear  that  if 
w  =  the  number  of  feet  in  the  width  of  the  border 
cut,  then  the  dimensions  of  the  uncut  part  of  the 
lawn  are  (60  -  2  ^/?)  and  (80  -  2  w?). 

Hence,  (60  -  2  lo)  (80  -  2  w?)  =  ^  •  60  •  80. 

Complete  the  solution. 

25.  A  farmer  is  plowing  a  field  whose  dimensions  are  40 
rods  and  90  rods.  How  wide  a  border  must  he  plow  around 
the  field  in  order  to  have  completed  |  of  his  plowing  ? 

26.  The  numerator  of  a  certain  fraction  is  8  less  than  the 
denominator.  If  the  denominator  and  numerator  each  be 
increased  by  5,  the  resulting  fraction  is  twice  the  fraction 
obtained  by  increasing  the  original  denominator  by  1.  Find 
the  fraction. 

27.  An  automobile  made  a  trip  of  50  miles,  10  miles  within 
the  city  limits  and  40  miles  outside  the  city  limits.  Outside 
of  the  city,  the  rate  was  increased  15  miles  an  hour.  If  the 
trip  took  2|  hours,  find  the  rate  at  which  they  travelled  within 
and  outside  of  the  city  limits. 


w  'w 

wlw 

' 

80-2W 

1 J 

CM 

|S 

W_" 

Iw 

--w"!w- 

QUADRATIC   EQUATIONS  2T5 

28.  The  numerator  of  a  certain  fraction  exceeds  its  denom- 
inator by  5.  If  the  numerator  be  decreased  by  3  and  the  de- 
nominator be  increased  by  4,  the  sum  of  the  new  fraction  and 
the  original  fraction  is  3.     Find  the  original  fraction. 

29.  A  bicyclist  rides  a  number  of  hours  at  a  number  of 
miles  an  hour  which  exceeds  the  number  of  hours  by  3 ;  an 
automobilist,  starting  3  hours  after  him,  overtakes  him  by 
going  two  and  one  half  times  as  fast  as  he  did.  Find  the  rate 
of  each. 

30.  The  circumference  of  the  fore  wheel  of  a  carriage  is  less 
by  4  feet  than  the  circumference  of  the  hind  wheel.  In 
travelling  1200  feet,  the  fore  wheel  makes  25  revolutions  more 
than  the  hind  wheel.     Find  the  circumference  of  each  wheel. 

199.  Quadratic  Equations  having  Two  Unknowns. 

Example  1.   Find  two  integral  numbers  the  sum  of  whose 

squares   is   90,  and  such   that   the   larger  exceeds  twice  the 
smaller  by  3. 

Solution  :  1.   Let  I  =  the  larger  number, 

and  8  =  the  smaller  number. 

2.  Then  1^  +  8^  =  90.  (1) 
and  1-2  8  =  3.                                             (2) 

3.  Eliminate  I  by  the  substitution  method  (§  168). 
From  (2)  l={2s  +  S). 
Substitute  in  (1):  .-.  (2  s  +  3)2  +  s^  =  90. 

.-.  4  s2  4.  12  s  +  9  4-  s^  -  90  =  0,  or  6  s2  +  12  s  -  81  =  0. 
Factoring :  (5  s  +  27) (a  -  3)  =  0.     .-.  s  =  3,  or  -  V. 

Check  :  If  s  =  3,  Z  =  9  ;  3^  +  92  =  9  +  81  =  90. 
—  ^  cannot  be  a  solution,  as  integral  numbers  were  desired. 

As  in  simultaneous  linear  equations,  when  there  are  two 
unknowns,  two  equations  must  be  given. 

These  two  equations  may  both  be  quadratic  equations,  or 
one  may  be  a  quadratic  and  the  other  a  linear  equation  as  in 
the  example  above.     In  this  course,  only  the  combination  of 


276  ALGEBRA 

one    quadratic    equation    and   one    linear   equation   will    be 
given. 

The   substitution   method   of    eliminating   one   variable   is 
usually  the  most  convenient. 

Example  2.    Solve  the  pair  of  equations  ~    '         \A 

[xy  =  —  15.       (2) 

Solution  :  1.   Solve  (1)  fora; :  x  =  (2  —  y). 

2.  Substitute  in  (2) :        y(2-y)  =  -  15.  (3) 

3.  .-.  2y- 2/2+ 15  =  0,  or  y2_2y- 15=0.  (4) 

4.  ...  (y_  5)(y +  3)=:0.    .-.?/  =  + 5  and  2/ =-3. 

5.  Since         x  =  2  —  y;  when  y  =  6,  a;  =  2  —  5  =  —  3; 

when  y  =  _3,  x  =  2-(-3)  =  +  5. 
First  solution  :  oj  =  —  3,  y  =  -f  5. 

Second  solution  :  x  =+  6,  y  =  —  3. 

Check  :  The  solutions  may  both  be  checked  by  substitution. 

EXERCISE   125 

Solve  the  following  pairs  of  equations : 

fa;-f2/  =  —  3.  ^     [m^-f-  mn  —  n^  =  — 19. 


'    [xy  =  —  54:.  "    i  m  —  n  =  —  7. 

2     ^a^  +  6^  =  113.  g     ix-y  =  -i. 

a  —  b  =  —  l.  '    [xy  =  4:5. 

5x'-3y^  =  -7.  fy-2x=10, 

y  +  2x  =  l.  i3a^-32/'  =  -5. 

27'2_rs=6s.  f?.?^^ 

•    V  +  2s  =  7.  10.      a      6~5* 

g       2y  +  2x  =  hxy.  [a  +  6  =  16. 

2x  +  2y  =  6.  f    18          14   ^g 

3c  +  2d  =  -2.  11-    \r-s     r  +  s 

cd  +  8c  =  4.  [r-2s  =  l. 

12.   Find   two  numbers  whose  sum  is   15  and  the  sum  of 
whose  squares  is  113. 


QUADRATIC   EQUATIONS  .  277 

13.  Find  two  numbers  whose  difference  is  9  and  the  sum  of 
whose  squares  is  221. 

14.  Find  two  numbers  whose  difference  is  7,  and  whose  sum 
multiplied  by  the  greater  gives  400. 

15.  Find  two  numbers  whose  difference  is  4,  and  the  sum  of 
whose  reciprocals  is  |. 

16.  Find  the  number  of  two  digits  in  which  the  units'  digit 
exceeds  the  tens'  digit  by  2,  and  such  that  the  product  of  the 
number  and  its  tens'  digit  is  105.     (See  §  177.) 

17.  The  sum  of  the  squares  of  the  two  digits  of  a  number 
is  58.  If  36  be  subtracted  from  the  number,  the  digits  of  the 
remainder  are  the  digits  of  the  original  number  in  reverse 
order.     Find  the  number. 

18.  The  area  of  a  rectangular  field  is  216  square  rods,  and 
its  perimeter  is  60  rods.  Find  the  length  and  width  of  the 
field. 

19.  A  and  B  working  together  can  do  a  piece  of  work  in  6 
days.  It  takes  B  5  days  more  than  A  to  do  the  work.  Find 
the  number  of  days  it  will  take  each  to  do  the  work  alone. 

20.  The  difference  in  the  rates  of  a  passenger  train  and  a 
freight  train  is  10  miles  per  hour.  The  passenger  train  requires 
1  hour  more  for  a  trip  of  175  miles  than  the  freight  train  re- 
quires for  a  trip  of  100  miles.     Find  the  rate  of  each. 

21.  The  altitude  of  a  certain  rectangle  is  2  feet  more  than 
the  side  of  a  certain  square ;  the  perimeter  of  the  rectangle  is 
7  times  the  side  of  the  square,  and  the  area  of  the  rectangle 
exceeds  twice  the  area  of  the  square  by  32  feet.  Find  the  side 
of  the  square  and  the  base  of  the  rectangle. 

22.  Find  the  sides  of  a  parallelogram  if  the  perimeter  is  24 
inches  and  the  sum  of  the  squares  of  the  number  of  inches  in 
the  long  and  short  sides  is  80. 

23.  One  of  two  angles  exceeds  the  other  by  5°.  If  each  is 
multiplied  by  its  supplement,  the  product  obtained  from  the 


278  ALGEBRA 

larger  of  the  given  angles  exceeds  the  other  product  by  the 
square  of  the  smaller  of  the  given  angles.     Find  the  angles. 

24.  Two  angles  are  supplementary.  The  square  of  the 
number  of  degrees  in  the  larger  angle  exceeds  by  4400  the 
product  of  the  number  of  degrees  in  one  angle  by  the  number 
in  the  other  angle.     Find  the  number  of  degrees  in  each  angle. 

25.  A  man  has  two  square  lots  of  unequal  size,  together 
containing  74  square  rods.  If  the  lots  were  side  by  side,  it 
would  require  38  rods  of  fence  to  surround  them  in  a  single 
inclosure  of  six  sides.     Find  the  length  of  the  side  of  each. 

IMAGINARY  ROOTS   IN  A  QUADRATIC  EQUATION 

200.   Example.     Solve  the  equation  x^  —  2  x  -[-  5  =  0. 
Solution  :  1.   Use  the  formula  method  of  solving  the  equation. 
a  =  l,  6  =  -  2,  c  =  5. 


2     ^  ^  -  5  rfc  V62  -  4  gc  ^  -f  2  ::b  V4  -  4  •  1  •  5 
2a  2 


The  question  arises  what  does  V  —  16  mean  ?  Is  —  4  the 
square  root  of  -  16  ?  No,  for  (  _  4)^=  +  16.  Is  +  4  ?  No, 
for  (  +  4)^  =  -f  16.  Thus,  no  number  with  which  the  student 
is  familiar  will  produce  —  16,  when  it  is  squared. 

201.  No  number  raised  to  an  even  power  will  produce  a 
negative  result ;  hence  an  even  root  of  a  negative  number  is 
impossible  up  to  this  point.  To  avoid  this  difficulty,  a  new 
kind  of  number  is  introduced. 

An  Imaginary  Number  is  an  indicated  square  root  of  a  negative 
number;  asV— 16;  V  — 3;  V— a^. 

The  numbers  previously  studied,  rational  and  irrational,  are 
called  Real  Numbers. 


QUADRATIC   EQUATIONS  279 

202.    Every   imaginary    number   can    be    expressed   as   the 
product  of  a  real  number  and  V—  1. 

V  — 1  is  indicated  by  /,  and  is  called  the  Imaginary  Unit. 


Thus,  V- 1«=  \/l(>(-l)=±     4V  -  1     =±4i. 


V-a^=  Va-^(-l)=±     aV-l    =±ai. 
v/^^5  =  VS  (  -  1)  =  ±  \/5  ■  y/^^  -  ±  iV5. 

HisTOKicAL  Note.     The  symbol  i  for  V—  1  was  introduced  by  Euler, 
one  of  the  greatest  mathematicians  of  the  eighteenth  centuiy. 

EXERCISE   126 
Express  the  following  in  terms  of  i : 


1.    V-49.      2.    V-:31>.        3.    V-100.      4.    V-81al 


5.    V-25.      6.    V-B4.        7.    V-144.      8.   V -121  a^b^. 
9.    V^      10.    V-^         11.    V^^.      12.    V^=^. 
13.    V^^.      14.    V-32.      15.    V^T8.       16.    V^^^T27 


17.    V-5.      18.    V-27.      19.    V-12.      20.   V-28. 


21.    Simplify  V-^. 


■■■H- 


22.    \/--.  25.    a/-".  28 


25. 

i- 

8 
"9* 

26. 

i- 

75^ 
'4  * 

27. 

<-- 

20 

23.    \/-|-  26.    \/-'^.  29 

36 


24.    Xl-:^'  27.    V'-^-  30 


:i  =  ± 

3tV3 

2 

.V: 

32 
"25* 

'■V- 

125 
64 

-V^ 

63 
lOO' 

280  ALGEBRA 

203.   Addition  and  Subtraction  of  Imaginary  Numbers. 
EXERCISE  127 


1.   Add  V^^  and  V-36. 

Solution  :    V—  4  +  V—  36  =  2i-\-Gi  =  8i. 

Note.  While  every  imaginary  number,  like  \/—  4,  has  two  values, 
cue  positive  and  one  negative,  in  i^roblenis  such  as  the  one  in  this  exercise, 
only  the  principal  root,  the  positive  one,  is  used,  as  in  the  case  of  surds 
(§  185). 


2.    V-16-hV-25. 


3. 


6.    V-1+V-49-V-64. 


7.    V--36  4-V-100-V-81. 


4.    V-81-V-(54. 


8.    V-a2_V_4«2^y_9^.^2^ 


5.    V-100  +  V-l()9. 


10. 


3  +  V-2^ 


9.    V-lGx^ 
13.    V^28- 


V-25a;2  +  V-49a;2. 


11.    V-18-V-8. 


14.    V-24-V^^r54 


7H-V-63. 


12.    V^^^  +  V^45.        15.    V-44-f  2V-11-V-99. 


16.    Simplify  4-|±V-f 


Solution  :  1.    -± 


V- 


27  _  5      Vir27^5     3v/3-  VC-l) 
4 "2  2  2  2 

^n      3rV| 
2  2 

_5±3i\/3 


The  numbers  in  Examples  1-15  are  called  Pure  Imaginaries, 
The  sum,  or  difference,  of  a  pure  imaginary  and  a  real  number, 
§  201,  as  in  this  exercise,  is  called  a  Complex  Number. 


QUADRATIC 

EQUATIONS 

1 
Simplify  the  following : 

.'■  i-H- 

» h^- 

u.  iW-l- 

"'■  io*>/    ™ 

"  IW-fe- 

-  il^xFl 

281 


204.   Meaning  of  Imaginary  Roots  of  a  Quadratic  on  the  Graph. 

Example.     Consider  the  equation  x^-^  x-{-2  =  0. 
Solution  :  1.    Solve  the  equation  by  the  formula  : 
a  =  l;  6  =  1;  c  =  2. 


Xi  = 


l+iV7. 


X2 


-l-^V-7^-l-fct  V7 

2  2 

-  1  -  I  V7 


2  2 

2.   Solve  the  equation  graphically.     (Review  rule  §  193.) 
Let  y  =  x2  +  x  +  2  : 


When  X  = 

0 

+  1 

+  2 

+  3 

-1 

-2 

-3 

-4 

then     y  = 

4-2 

+  4 

+  8 

+  14 

+  2 

+  4 

+  8 

+  14 

3.  The  graph  has  the  same 
shape  as  the  graphs  obtained 
when  solving  other  quadratic 
equations  ;  but  the  graph  does 
not  cross  the  horizontal  axis 
at  all.  Hence,  y  or  x^  +  x  +  2 
is  never  zero  for  any  real  value 
of  X. 

This  is  charaQteristic  of 
the  graph  of  a  quadratic 
which  has  imaginary 
roots. 


h 


4'--3- 


♦K) 


♦2_+3. 


t 


.t$_  +5. 


282  ALGEBRA 

V 

EXERCISE   128 

Solve  the  following  equations.    Express  the  roots  in  simplest 
form.     Draw  the  graphs  for  the  first  three  equations. 

1.  a;2  +  2.T-|-3  =  0. 

2.  x2  +  3x-|-4  =  0. 

3.  2x''-x-\-2  =  0. 

4.  3m2-2m  +  5  =  0. 

5.  5c2-7c  +  3  =  0. 

6.  St'  =  9t--5. 

7.  ll?-24-6  =  15r. 


9. 

2 
x-\-S  ' 

3a; 
5 

=  0. 

10. 

2a-  +  3 

_x 

X 

+  7 

11. 

30^1 

2  "^a; 

-1 

=  0. 

12.  2a;2-5aa;  +  7a2  =  0. 

13.  Ss^-4.xw-^2w^=:0. 


^_3^     1_^  ^ 

5      10     2       *  14.   5x^-7  xt-\-3t^  =  0. 


XVI.   SPECIAL   PRODUCTS  AND   FACTORING 

ADVANCED  TOPICS 

205.  In  paragraph  98  is  the  rule  :  "  The  product  of  the  sum 
and  the  difference  of  any  two  numbers  equals  the  difference  of 
their  squares  "  ;  thus,  {x  +  y){x  —  y)  =  x^  —  y^  for  all  numbers 
X  and  y. 

lix  =  2a  and  t/  =  3  6,  (2  a  +  3  6)(2  a  -  3  6)  =  4  a2  -  9  62. 

If  a;  =  14  and  y  =  5,  (14  +  5)(14  -  5)  =  169  -  25  =  144. 

If  a;  =  (a  4-  b)  and  y  =  {c  -\-  d),  then  similarly 

[(a  +  &)  +  (c  +  (?)][(«  +  h)  -  (c  +  d)^  =  (a  +  6)2  _  (c  +  dy. 

Likewise,  in  any  of  the  type  forms  studied  in  chapter  VIII, 
the  numbers  may  be  general  number  expressions. 

Example  1.    Multiply  (a  +  6  +  c)  by  («  +  6  —  c). 
Solution  :  1.   (a  +  6  +  c)(a  +  6  —  c)  =  {(«  +  6)  +  c}{{a  +  6)  —  c} 

=  (a  +  6)2  _  c2  =  a2  +  2  a6  +  62  -  c\ 
Here  x  =  {a  -\- h)  and  y  =  c. 

Example  2.    Multiply  {r  -^  s  -\- 1  —  n)hy  {r  -\-  s  —  t  -\-  n). 
Solution  :  1.  (r  j^  s  -\- 1  —  n){r  -{-  s  —  t  -\-  n) 

=  {(r  -hs)  +  (t-  n)}{(r  +  s)  -  (t  -  n)}  =  (r  +  s)2_  (t  -  ny 

=  r2  +  2  rs  +  s2  _  ^2  4.  2  «n  -  n2. 
Here,  x  =  (r  +  s)  and  y  =  (t  —  n). 

Note.  In  such  examples,  the  rules  for  introducing  parentheses  (§  .50) 
are  used.  The  various  terms  of  the  expressions  may  be  rearranged,  if 
necessary,  so  that  one  factor  becomes  the  sum  and  the  other  the  difference 
of  the  same  two  numbers,  when  the  terms  are  grouped. 

283 


284  ALGEBRA 

EXERCISE  129 
Find  the  following  products  mentally  : 

1.  S(a-\-b)+5ll(a  +  b)-5l. 

2.  l(m-^n)-2pl{{m-\-n)-\-2pl. 

3.  \10-(r-^s)lllO-\-{r  +  s)\. 

4.  sSp-{c+d)l\S2i-{-(c-{-d)l. 

5.  \(c  +  2  d)  -  11  all(c  + 2  d)-h  Hal. 

6.  (a  —  b  +  c)(a  —  6  —  c). 

7.  (x-y -{-z)(x-y -z). 

8.  (a'  +  a  -  l)(a2  -  a  +  1). 

9.  (a^  -f  a6  4-  ?>')(a2  -  ah  +  6^). 

10.  (a -{-2b -3  c)(a  _  2  5  +  3  c). 

11.  (3a;  +  42/  +  22)(3aj-42/-22;). 

12.  (a^  +  aj- 2)(a^-a;-2). 

13.  (a  +  r  —  c  +  f^)  (a-{-r-{-  c  —  d). 

14.  (a  —  6  +  m  -{-  7^)  (a  —  b  —  m  —  n). 

15.  (2  £c  4-  2;  —  ?/  +  w)(2  X  —  z  —  y  —  ^'). 

16.  J(«  +  ^)  +  2(a  -  6)}  J(a  +  &)  -  3(a  -  b)}. 
Solution  :  Just  as  (x  -\-2y)(x—Zy)  =  x^  —  xy  —  6y^ 

so  {(a  +  6) +2  (a -&)}{(«  +  &) -3  (a -ft)} 

=  (a +  6)2-  (rt  +  6)(a_6)  _6((i-6)2 
=  (a2  +  2  a&  +  &2)  _ (^2 _  ft-2)_  6(a2  _  2  a&  +  ft^) 
=  a2  +  2  a&  4-  &2-  a2  _{_  ?^2  _  0 a2  _^  12  a6  -  6  62 
=  Uab-6a'^-4b^. 
Note,    ic  =  (a  +  6)  and  y  =  (a  —  6). 

17.  J(^^^  +  *0- 4H0^  +  w)- 5J. 

18.  ;(a;_iy)  +  8n(aj-?/)-6(. 

19.  {Sx-(y-hz)il2x-(y-{-z)l. 


SPECIAL   PRODUCTS  AND  FACTORING  285 

20.  \x-}-3y-^15zl\x-\-3y-10z\. 

21.  lr  +  2s-3tl]r  +  2s-^7t\. 

22.  \Sp-4{q-\-r)\\4.p-5((j-{-r)l. 

23.  \^-\-2x-hl\lx'-{-2x-5l. 

24.  [(a  +  b)  -  5Y.  28.  [ft  -\-3b-  cy. 

25.  [0+(m-n)]'-  29.  [a  -  6  +  c  -  d]". 

26.  [2rt-(c  +  d)]2.  30.  [2r  +  s-«H-a;]'- 

27.  [a  +  6  +  c]2.  31.  [3a-6  +  2c-dJ. 

206.    General  Problems  in  Factoring. 
Example  1.   Just  as  scF—if  =  {x-{-  y){x  —  y) 

so  (m  -  nf  -  25  a'  =  \{m  -  n)+ba\  \{m-n)-6a\. 

Example  2.    Just  as  ar^  -  3  a;  —  88  =  (a;  —  11) (a;  +  8) 

%o{a-2hf-3{a-2h)-%^=\(a-2h)-ll\\{a-2h)  +  ^\. 

=  (a-26-ll)(a-26  +  8). 
Note,     x  is  (a  —  2  6). 

EXERCISE  130 
Factor  completely  the  following  expressions : 

1.  (a  +  6)2_c2.  11.    {x'-4.y-{x  +  2)\ 

2.  {m-nY-x'.  12.    9(m-r2)2- 12(m -7i)+4. 

3.  ar^__(y_^.2;)2.  13.    (a;  -  2/)2  -  (m  -  ?i)l 

4.  m2-(7i-i))2.  14.   (a2-2a)^+2(a2-2a)+l. 

5.  ilx-2yy-y\  15.    (l+w')'-4«l 

6.  (a  +  6)2-|-23(a-f6)-f  00.        16.    (ar'+3  a;)^  4-4(a;24-3a;)  +  4. 

7.  (a;_2/)2  +  2(a;-?/)-63.  17.    (9  a^  +  4/- 144  al 

8.  (a;4.2/)2-5(aj  +  2/)-36.  18.    (a2-|-7a)H20(aH7«)-96. 

9.  {r-\-8f+4:{r  +  s)t-^f.        19.    (m  +  w)2  4-7(7?i4-rO- 144. 
10.    (j9-g)2-i-8(/>~9)y-20  7^.      20.    (a;^ ^ ^j _  9^2 „  9^ 


286  ALGEBRA 

21.  (x-\-yy-z\  26.  (x-^yy-\-(x-y)\ 

22.  (r-\-sy-{-8t^  27.  ^x^-ix"" -{-ly. 

23.  {m  +  7if-(m-ny.  28.  27  m^ -(m -nf. 

24.  a3+(a  +  l)^  29.  (2  a- 6)«- (a  + 2  ?))^ 

25.  a3-8(a  +  6)^  30.  {x  +  Syy- {x-3  yy. 

FACTORING  POLYNOMIALS 
207.   Polynomials  Reducible  to  the  Difference  of  Two  Squares. 

Certain  polynomials  may  be  put  into  the  form  of  the  differ- 
ence of  two  squares  by  grouping  certain  terms. 

Example  1.    Factor  2  mn  -\-m^  —  l-\-  v?. 

Solution  :  L    2  mn  +  w^  —  1  +  ti^  =  (m^  +  2  mn  +  n^)—  1. 

=  (m  4-  w)2  -  1. 

=  (w  +  n  +  l)(m  +  w  -  1).  (§206) 

Example  2.   Factor  o?  -  (?  ^h^  ^d? -2  cd-  2  ah. 
Solution  :  1.    a^  -  c^  +  IP-  -d:^-2cd-2ah 

=  (a2  -2ab-\-  b'^)-(c^  +  '2cd  +  cP) 

=  (a_6)2_(c  +  (Z)2 

=  {(«-  6)  +  (c  4-  d)]{(a  -  b)-(c  +  d)} 
=  (^a  —  b  +  c-\-d)(a  —  b  —  c  —  d). 

EXERCISE  131 
Factor : 

1.  a^-2ab-\-b^-c^,  6.  2  mn-n^-^1 -m\ 

2.  7rv'-^2m7i  +  n^-p\  7.  9a2-24a6-f  16&2_4c2. 

3.  a^-x^^2xy-y\  8.  16ar^-4?/2  4.202/:3-2522. 

4.  a^  —  i/'^  —  2^  +  2  2/2.  9.  4: 71^  -\- m^  —  x^  —  4:  mn. 

5.  52_44.2a6  +  a2.  lo.  4^2-66-9-62. 

11.  10xy-9z^-^y^-^25x\ 

12.  a2-2a6  +  62-c24-2cd-d2. 

13.  a^  —  b'^  +  x"  —  y^-{^2ax-^2by. 


SPECIAL   PRODUCTS   AND   FACTORING  287 

14.  J^  -(-  m^  —  if  —  ir  —  2  mx  —  2  ny. 

15.  2xij-a'  +  x^-2ab-b--\-if. 

16.  4a2  +  4rt6  +  ^/-9c^  +  12c-4. 

17.  16y'-3(^-Sxi/-z^  +  x'-12z. 

18.  m^  -  9  n^  +  25  a^  -  b-  -  10  cwi  +  6  6m. 

19.  4a2-c2-12a6  +  2crf  +  962_d2. 

20.  9x*-4:X^-\-z^-6x'z-20x7j-2ri!/'. 

208.    Certain  polynomials  can  be  factoreil  by  (jronpiny  their 
terms. 

Type  Forvi :  ab  +  ac  +  bd -\-  cd  =  {a -\-  d){b  -\-  c). 

Example  1.     Just  as  ax  -\-bx  =  (a-\-  b)x  (§  34) 

so,  a(x  +  2/)  +  b{x  +  y)  =  (a-^b)(x-\-  y)  (§  34) 

Example  2.     Factor  6  a;^  -  15  a^  -  8  a;  -f  20. 

Solution:  1.    Qx'^-\bx^-^x-\-20={(d  j?-\^x^)-{'^x~2Q)        (§47) 

=  3x2(2x-5)-4(2x-5)  (§94) 
=  (3x2_4)(2x-5). 

EXERCISE  132 
Factor  < 

1.  2a(a;  +  2/)-3(.r4-2/).  H.  2 +  3  a-8  a^- 12a^ 

2.  5  m(r -h  s)  +  2  n(r  4- .*J).  12.  3  ar^  +  6  a^  +  a;  +  2. 

3.  Sp(2x  —  y)  —  r(2x-y).  13.  lOma;— 15  >ia;- 2m+3n. 

4.  8(f  + 1«)  —  wi(^  4-  w).  14.  a'^a;  +  atca;  —  a^by  —  bhy. 

5.  a{h-\-c)  —  dih  +  c).  15.  a^bc  —  cw^d -\- abhl  —  bccE 

6.  a6  4- a?t  +  67M  +  m7i.  16.  30  a'^  —  12  a^  _  55  a  +  22. 

7.  ax-ay +  bx-by.  17.  56  -  32  a;  4- 21  a.-^  -  12  arl 

8.  ac  —  ad  —  6c  4-  6d.  18.  3  aa;  —  a?/  4-  9  6a;  —  3  by. 

9.  a«  +  a2  4-a  +  l.  19.  4  ar' 4- ar^/ -  4  /  -  16  a;?/. 
10.  4ar'  — 5»*  — 4  a;  — 5-  20.  rt  —  rn  —  sn -\- st. 


288  ALGEBRA 

21.  ar -[- as -^  br -{- hs  —  cr  —  C3. 

22.  ax-\-a}j  ~  az  +  bx  —  bz-^  by. 

23.  am  —  bm  —  cp  -\-  ap  —  mi  —  bp. 

24.  x^  —  ic^;-  +  a;^?/  +  xy'^  +  ^/"^  —  yz'^. 

25.  2  ax  +  ex  -i-  S  by  —  2  ay  —  3  bx  —  cy. 

209.  In  the  following  list  of  examples,  the  types  of  factor- 
ing stndied  in  this  chapter  will  be  used.  Before  taking  up 
the  list  of  examples,  review,  if  necessary,  the  rules  for  obtain- 
ing the  H.  C.F.  of  .two  or  more  expressions  in  chapter  IX  and 
the  rules  for  operations  with  fractions  in  chapter  X. 

EXERCISE  133 
Find  the  H.  C.  F.  and  the  L.  CM.  of  the  following  : 

1.  3  a^  -  21  a^  -  a  +  7,  and  a^  +  6  a  -  91. 

2.  ac  +  ad  —be  —  bd,  and  a^  —  6  ab  -\-  i)  b"^. 

3.  a^-\-b^-c'  +  2  ab,  and  a^  -b~-c^-\-2  be. 

4.  m^  -  4  m,  m^  +  9  m^  —  22  m,  2  m"^  —  4  m^  _  3  m^  +  6  m. 

5.  3a^-a'b-\-Sab-  b\  27  a^  -  b\  9  a^  -  6  «?>  +  b\ 

6.  16  mi'^  —  71'^,  16  711^  —  8  771^71^  +  n*,  2  7nx  +  2  my  —  7ix  —  ny. 

7.  a^  —  d~x  —  av?  +  ar^,  3  a^  —  3  a^x  -f  5  aa^  —  5  a^. 

Reduce  the  following  fractions  to  lowest  terms : 

x^  —  y^  -\- z^ -\- 2  xz  ^    ax  —  bx  —  ay  -f  by 

'   d?  —  y^  —  z^  -\-2yz  a? —  V^ 

4m-^-10?yi^-6m+15       ^^    2ae-2bc- ad-^bd 
6m'^  +  8m=^-9m-12*         *  d^-^(? 

Simplify  the  following : 


13. 


x^  -\-  y''^  —  z^  —  2  xy  x^  —  y'^  —  2  yz 
a^  +  &-^     f.  ab  +  c2 


64.C  V         a'-ab  +  b'- 


SPECIAL   PRODUCTS  AND   FACTORING  289 

a^—h-  —  c^  —  2hc  .  a  —  h  —  c 
rt2_52_^_,.2^,c  *  a  +  h-c 

ar  —  as-{-  br  —bs  ar  +  as  4-  ^^f  +  ^^ 
r^-^  '    a^^2ab-^b' 

16.  Solve  the  equation :  x^+ax  —  3ab  —  3  bx  =  0. 
Solution  :  1.   a;'^  +  aa:  —  3  a6  —  3  hx  =  0. 

2.  Factoring:  ic(x  + a)- 3  &(»;  + rt)  =  0. 

(x-3&)(x  +  a)  =  0. 

3.  .-.  a:  =  3  Z>  or  a;  =  -  a.  (§  110) 

Check  :  When  x  =  8  6,  does  (3  6)*  +  a(3  6)  -  8  aft  -  3  6(3  6)  =  0  ?,     , 
Does  0  62  4.  3  a?^  _  3  a?,  _  9  ?;2  =  0  ?     Yes. 
Whenx=-  a,  does  (- a)2  +  a(- a)- 3  a6  -  36(- a)  =0? 
Does  a2  -  a2  -  3  a6  +  3  a6  =  0  ?     Yes. 

17.  x^  -\-ax  —  ab  —  bx=0.          19.    ax'^ —  bx— acx-\-bc  =  (). 

18.  ar^  — 2?w7i-m-— ?r  =  0.      20.    ax- —2  dx-Q>d-\-3  ax=0. 


XVII.    RATIO,   PROPORTION,   AND   VARIATION 

210.  The  Ratio  of  one  number  to  another  is  the  quotient  of 
the  first  divided  by  the  second. 

Thus,  the  ratio  of  a  to  6  is  -;  it  is  also  written  a  :  h.      The 

h 

numerator   is   called   the  Antecedent   and  the  denominator  is 

called  the  Consequent. 

All  ratios  are  fractions  and  are  subject  to  the  usual  rules  for 

operations  Avith  fractions. 

211.  The  ratio  of  two  concrete  quantities  may  be  found  if 
they  are  of  the  same  kind  and  are  measured  in  terms  of  the 
same  unit. 

Thus,  the  ratio  of  3  lb.  to  2  lb.  is  |  ;  and  the  ratio  of  350  lb.  to  2  tons  is 
?¥A  or  A. 

EXERCISE   134 

Express  the  following  ratios  and  simplify  them : 

1.  3  to  9.       3.    5.Tto2a;.  5.    |  to  yV      7.   25  to  375. 

2.  12  to  2.     4.    6anol5al     6.    y\  to  i       8.    d'-hHo  o?-h\ 

9.    A  line  15  inches  long  is  divided  into  two  parts  which 
have  the  ratio  2 :  3.     Find  the  parts. 
Solution:  1.   Let  x  —  the  short  part. 

2.  Then  15  —  a;  =  the  long  part. 

3.  Then  -^^_  =  ?. 

15 -X     3 

Complete  the  solution. 

10.  Divide  a  line  63  inches  long  into  two  parts  v/hose  ratio 
is  3  :  4. 

290 


RATIO,    PROrORTTOX,    AND   VARTATTON  291 

11.  Divide  36  into  two  parts  such  that  the  ratio  of  the 
greater  diminished  by  4  to  the  less  increased  by  3  sliall  be  3 :  2. 

12.  The  ratio  of  the  height  of  a  tree  to  the  length  of  its  shadow 
on  the  ground  is  17  :  20.  Find  the  height  of  the  tree  if  the 
length  of  the  shadow  is  110  feet. 

13.  Divide  99  into  three  parts  which  are  as  2  :  3  :  4. 
Hint  :  Let  the  parts  be  2  a;,  3  a;,  and  4  x. 

14.  Divide  a  farm  consisting  of  720  acres  into  parts  which 
are  as  3  :  5. 

15.  Divide  $  1000  into  3  parts  which  are  as  5  : 3  :  2. 

212.  A  Proportion  is  a  statement  that  two  ratios  are  equal. 
The  statement  that  the  ratio  of  a  to  6  is  equal  to  the  r^tio  of 
c  to  d  is  written  either 

—  =  -,   or  a:  0  =  c  :  (6. 
h     d 

This  proportion  is  read  "  a  is  to  6  as  c  is  to  d.'' 

Thus  3,  9,  5  and  15  form  a  proportion  since  |  =  ^^3^. 

HisTORicAF.  Note.  Leibnitz,  1646-1716,  was  instrumental  in  estab- 
lisliing  the  use  of  the  form  a  :  b  =  c  :  d. 

213.  The  first  and  fourth  terms  of  a  proportion  are  called  the 
Extremes,  and  the  second  and  third  the  Means. 

In  the  proportion  a:  b  =  c:  d,  a  and  d  are  the  extremes,  and 
b  and  c  are  the  means  ;  a  and  c  are  the  antecedents,  and  b  and 
d  are  the  consequents. 

EXERCISE   135 

Find  the  value  of  the  literal  number  in  the  first  six  of  the 
following  exercises  and  of  x  in  the  remaining  ones  : 

1     -  =  A  3    Jl=i:  5    ^  —  x_^ 

'   3     27'  *   16     5*  3     ~2* 


< 
16^ 

c 
"5 

9 
24' 

z 

2     ?=^^.  4      ^  =  ?  6     2— ?  =  5 

'   y      iO  '    24      «  "4  +  ^2' 


292 


7. 

a  _x 
b~~c 

8. 

a       X 

2b~3c 

ALGEBRA 

9.  ^  =  ?:. 

SX        t 

11. 

a  —  x     a 
X           b 

0.  ^  =  A. 

12. 

a         n 

np     nx 

x—m     X 

214.  A  Mean  Proportional  between  two  numbers  a  and  b  is 

the  number  x  in  the  proportion  a  :  x  =  x :  b. 

2      X 
A  mean  proportional  between  2  and  3  is  aj  in  :  -  =  -  • . 

X     3 

.-.  x2  =  6  ;  x=  ±  V6. 

Thus,  there  are  two  mean  proportionals  between  any  numbers.     Gener- 
ally the  positive  one  is  used. 

215.  The  Third  Proportional  to  two  numbers  a  and  b  is  the 
number  x  in  the  proportion  a :  b  =  b  :  x. 

2  3 
Thus,  the  third  proportional  to  2  and  3  is  x  in  :    -  =  - ; 

3  X 
.*.  2  ic  =  9  and  x  =  4.5. 

216.  The  Fourth  Proportional  to  three  numbers  a,  b^  and  c  is 

the  number  x  in  the  proportion  a:b  =  c:  x. 

2      4 

Thus,  the  fourth  proportional  to  2,  3,  and  4  is  the  number  x  in :  -  =  -< 

O  X    y 

.-.  2  a:  =  12  and  x  =  6. 

Note.    The  numbers  must  be  placed  in  the  proportion  in  the  order  in  which 
they  are  given,  as  in  the  illustrative  examples. 


EXERCISE  136 
Find  the  fourth  proportional  to : 

1.  2,5,  and  4.  4,  35,  20,  and  14. 

2.  5,  4,  and  2.  5.    6  a,  2  b,  and  c. 

3.  7,  3,  and  14.  6.   x,  y,  and  xy. 

Find  the  mean  proportionals  between : 

7.  18  and  50.  9.    2  a  and  a. 

8.  2^  and  f .  10.   12  m^w  and  3  mn^. 


RATIO,   PROPORTION,   AND   VARIATION  293 

11. —  and  — -^- — 12.    ar^  —  v^  and  — - — '^      ^  « 

a+4  a-f2  ^  x  —  y 

13-16.   Find  the  third  proportional  to  the  numbers  in  ex- 
amples 7,  8,  9,  and  10. 

17.  Find  the  third  proportional  to  a^  —  9  and  a  —  S. 

18.  Find  the  third  proportional  to  10  a;  and  3  y. 

19.  Find  the  fourth  proportional  to : 

2  a^  —  2  2/^     a^  —  y^         ^ax  —  by-\-ay  —  bx 
a-\-b     '    ^^^^^'   ^^        a^J^xy-^f       ' 

20.  Find  the  mean  proportionals  between : 

ax  —  ay  —  bx -\- by  a^  —  y^ 


x^  +  xy-\-y'^  (a  —  bf 

PROPERTIES  OF  PROPORTIONS 

217.  In  a  proportion,  the  product  of  the  means  is  equal  to  the 
product  of  the  extremes. 

This  property  of  a  proportion  is  proved  as  follows  : 

If  -  =  -,  then  ad  =  be,  by  clearing  of  fractions. 
b      d 

Example.     Since  §  =  |,  2  •  9  should  equal  3  •  (y.     Does  it  ? 

218.  If  the  product  of  two  numbers  is  equal  to  the  product  of 
two  other  numbers,  one  pair  may  be  made  the  means  and  the  other 
the  extremes  of  a  proportion. 

If  mn  =  xy,  then  ^  =  ^ . 

X      n 

Prove  this  by  dividing  both  members  of  the  given  equation 

by  nx. 

Prove  that  the  following  proportions  also  are  true : 

(a)  ^  =  ^   (divide  by  ny).  (ft)  -^  =  ^ .  (c)  -  =  -^  • 

^  ^    y      n   ^  ^     ^^  ^  ^   m      y  ^  ^    x      m 

ExAMPLK  1.     Since  3  •  8  =  6  •  4,  f  should  equal  |.     Does  it  ? 
Example  2.     Write  three  other  proportions  which  should  be  true  ac- 
cording to  the  property  given  in  this  paragraph. 


294  ALGEBRA     ' 

219.  In  any  i^roportion,  the  terms  are  in  proportion  by  Alter- 
nation ;  that  is,  the  first  term  is  to  the  third  as  the  second  is  to  the 
fourth.  f 

rx:  a      c  ah 

it  -  =  - ,  prove  -  =  - . 

b      d^  c      d 

Suggestion.     Use  §  217  and  then  divide  both  members  of  the  equation 

by  cd. 

Example.     Since  |  =  ^\,  then  |  should  equal  j%.    Does  it  ? 

220.  In  any  proportioii,  the  terms  are  in  proportion  by  Inver- 
sion ;  that  is,  the  second  term  is  to  the  first  as  the  foiirth  is  to  the 
third. 

TP  a      c  b      d 

If  -  =  -,  prove  _  =  - . 

b     d  a     c 

Suggestion.     Use  §  217,  and  then  divide  both  members  of  the  equation 

by  ac. 

Example.     Since  |  =  j%,  then  |  should  equal  \2,.    Does  it  ? 

221.  Bi  any  proportion,  the  terms  are  in  proportion  by  Compo- 
sition ;  that  is,  the  sum  of  the  first  tivo  terms  is  to  the  second  as 
the  sum  of  the  last  two  terms  is  to  the  fourth. 

yp  a      c  a-\-b      c-\-  d 

If  -  =  - ,  prove  ^    =    ^     - 

b     d^  b  d 

Suggestion.    Add  1  to  both  members  of  the  given  equation. 

24  24-6  44- 12 

Example.     Since  -  =  — ,  then      "^     should  equal  — ^^^^ — .     Does  it  ? 
6      12  6  ^12 

222.  In  any  proportion,  the  terms  are  in  proportion  by  Divi- 
sion ;  that  is,  the  difference  of  the  first  two  terms  is  to  the  second, 
as  the  difference  of  the  last  two  is  to  the  fourth. 

xnttc  a  —  b      c  —  d 

It  -  =  - ,  prove  = • 

b      d'  ^  b  d 

Suggestion.     Subtract  1  from  both  members  of  the  equation. 

Example.     Since  —  =  ^,  then  ^^-=^  should  equal  i^^  •    Does  it  ? 
2       8  2  ^8 


RATIO,   PROPORTION,   AND  VARIATION  295 

223.  //*  cm  1/  proportion,  the  terms  are  in  proportion  by  Compo- 
sition and  Division ;  that  /.s,  the  sum  of  the  first  two  terms  is  to 
their  dijference  as  the  smn  of  the  last  two  terms  is  to  their 
difference. 

T*  «      c   _„^„^  a-\-h     c-\-d 

It -  =  -,  prove  — ——■= — ■ 

h     d  a—bc—d 

Proof.     1.   Since  ?=-,  then  ^^Jt^  =  !^-±^.  (Composition) 

b      d  b  d 

2.  Since  -  =  -,  then  '    ^Lsil  =  ^--A,  (Division) 

b     d  b  d  ^  ^ 

3.  Divide  the  members  of  the  equation  in  step  1  by  those  of  the  equa- 
tion in  step  2 :  ,   ,  7  ,   ^  i 

^  a  +  b  .  a  —  b_c  +  d  .  c  —  d 

b  b     ~     d  d    ' 

4.  Simplifying  step  3:  a±b^c±d 

a—  b     c  —  d 

Example.    Since  ^  =  ^ ,  then,  ^^-±^  should  equal  i^-±-^  •   Does  it  ? 
2       8  '10-2  ^       15-3 

224.  In  a  series  of  equal  ratios,  the  sum  of  the  antecedents  is  to 
the  sum  of  the  consequents  as  any  antecedent  is  to  its  consequent. 

yo  a      c      e      .  a  -\-  c  -\-  e  -\-  etc.      a 

If  -  =  -  =  -,  etc.,  prove  — — — — — — =  -  • 

b     d     f  ^  6  4-ci+/+etc.      b 

Proof.    1.  Let  v  =  the  common  value  of  the  equal  ratios  ^ ,  -,  - ,  etc. 

b    d    f 

2.  Then  since  -  =  v,  a  =  bv. 

b 

-  =  V,  c  =  dv. 
d 

-  =v,  e=fv. 

3.  Then  (a -\-  c  +  e)=  bv  +  dv  -\-fv  =  v{b  +  d  +/). 

4.  U(b+d+/) '     — ' ■ —  =  v.     .•. — ■ =      or  -    or  —  • 

b+d+f  h^d+f     b        d        f 

Example.    Since^-=-=A     J_±A±A  should  equal  ^ .    Does  it? 
2     6      10     2  +  6+  10  ^        2 

Historical  Note.  All  of  these  properties  of  a  proportion  were  known 
to  Euclid,  300  B.C. 


296:  ALGEBRA 

225.  There  are  several  other  properties  of  a  proportion 
which  follow  directly  from  properties  of  an  equation  or  of  a 
fraction.  . 

(a)  If  -  =  -    then  —  =  —  •  Raise  both  members  to  the  third 

b      d^  6^      #  power. 

(h^  Tf  -  —  -     thpTi  '^^  —  Extract  the    cube    root    of    both 

b~d'  \/?>  ~  \/d  members. 

Multiply  numerator  and  denomi- 
(c)   If  T  = -J  then  — -  =  -— •  nator  of  the  first  ratio  by  m,  and 


b      d  mb      nd 


(a)  If  -  =  -,  then  — -  =  — -•  -u    rn 

b      d  nb       nd  equation  by  — 


of  the  second  by  n. 

Multiply    both    members   of    the 


11 


226.  In  the  preceding  paragraphs,  some  of  the  simple  prop- 
erties of  a  proportion  have  been  given.  There  are  many  others 
which  may  be  derived  by  means  of  these  simple  properties. 

^  J.  a      c  2  a  4- 3  6      2a- 3  6 

Example.     If  -  =  -,  prove !—-—=- -— • 

h     d'  ^  2c  +  3d      2c-3d 

Proof.     1.    Since  ?  =  ^.  then  |^  =  |^;  •  (§  225,  d) 

b      d  6b      od 

2.  Then  2  a  +  36  ^  2  c  +  3d,  (By  §  223) 

2a-3&      2c-3d!  K   y  ^        J 

3.  Then  2  a -f  3  6  ^  2a_-3j,  (By  §  219) 

2c+3(Z      2c-3(Z  ^   '  ^        ^ 


EXERCISE   137 

1.  Write  by  inversion : 

/  \  3     15  ,,x    2     m  ,.   a     X 

(")  4  =  20-  ^*)  5=x-  ^"^  b  =  V 

2.  Write  these  same  three  proportions  by  alternation. 

3.  Write  these  same  three  proportions  by  composition. 

4.  Write  these  same  three  proportions  by  division. 

5.  Write  the  proportion  (c)  in  Example  1 : 

(a)  by  inversion  and  the  result  by  composition ; 


RATIO,   PROPORTION,   AND  VARIATION  297 

(&)  by  alternation  and  the  result  by  division ; 

(c)  by  composition  and  the  result  by  alternation  ; 

(d)  by  division  and  the  result  by  inversion. 


6. 

-r ,.  m      X               J.^    1.  VI  -\-n      n 
If  _  =  - ,  prove  that             =  -  • 
n      y                        x-^y      y 

7. 

If  -=z-,  prove  that  — ^-- =  ■          • 
s      b    ^                      r            a 

8. 

If  _  =  - ,  prove  that  =  -  • 

b      d^                   c~d      d 

9. 

If -=^,  prove  that      „.=      ,.• 

10. 

If  2_^,  prove  that  2«-36^2c-3cJ 

EXERCISE  138 
Proportion  in  Geometry 

1.  In  a  triangle  in  which  DE  is  parallel  to  BCj  m :  r 

To  test  this  truth  :  (a)  measure  m,  n,  r,  and 
s ;  (6)  find  the  value  of  the  ratio  m :  r  and  of 
n:s;  (c)  compare  these  two  ratios. 

*This  truth  may  be  tested  in  any  triangle.  It 
may  be  expressed  thus :  the  upper  segment  on 
one  side  is  to  the  lower  segment  on  that  side  as 
the  upper  segment  on  the  other  is  to  the  lower 
segment  on  the  other. 

2.  Write  the  proportion  —  =  -  by   alternation.      Express 

the  resulting  proportion  in  words  as  in  Example  1. 

3.  Write  the  proportion  of  Example  1  by  composition  and 
express  it  in  words. 

4.  Write  the  proportion  of  Example  1  by  inVBrsion  and 
express  it  in  words. 

6.   If  ^ID  =  7,  DB  =  4,  and  .AE  =  8,  find  EO. 


298 


ALGEBRA 


6.  If  AB  =  12,  AD  =  5,  and  AC  =  14,  find  AE. 
Hint.     Let  AE  =  x,  and  C^  =  14  -  x. 

7.  If  AD  =  Z>jB,  how  does  AE  comi)are  with  EO? 

8.  If  AD  =  20,  />B  =  8,  and  AC  = 
30,  find  AE  and  ^Xl 

9.  If  two  perpendicular  lines  BC 
and  DE  are  draAvn  from  one  side  of  an 
angle  to  the  other,  then  BC :  AC  = 
DE'.AE. 

Test  this  statement  by  measuring  the  lines  in  the  figure  and  finding  the 
value  of  the  ratios. 

10.  Draw  any  other  perpendicular  as  XY.  Find  the  ratio 
of  Xy  to  AY  and  compare  the  ratio  with  those  found  in 
Example  9.  What  do  you  conclude  about  all  ratios  obtained 
by  dividing  the  length  of  the  perpen- 
dicular by  the  distance  from  A  to 
the  foot  of  the  perpendicular  (like 
AY)? 

11.  Using  the  fact  stated  in  Exam- 
ple 9,  tell  how  to  find  the  height  of  the 
tree  in  the  ^gure,  if  the  height  of  the 
rod  and  the  lengths  on  the  shadows  of  the  tree  and  the  rod 

are  as  indicated. 

12.  Suppose  that  EF  and  AC  are 
perpendicular  to  DC  in  the  adjoining 
figure.  Suppose  that  EF  =  10  feet, 
OF  =  12  feet,  OC  =  150  feet,  and 
BC  =  20  feet.     Determine  AB. 

13.  Suppose  that  CD  and  AB  are 
perpendicular  to  AE  in  the  adjoining 
figure ;  that  AX  =  5  feet,  YB  =  ^ 
feet,  AE  =  im  feet,  CE  =  25  feet, 
and  CD  =  30  feet.     Find  XY 


RATIO,   PROPORTION,   AND   VARIATION  299 


VARIATION 

227.  Some  quantities  change  or  vary  and  are  called  Variable 
Quantities. 

Thus,  the  distance  between  a  moving  train  and  its  destination  varies,  — 
ihat  is,  it  decreases  ;  the  age  of  an  individual  varies  from  moment  to 
moment,  —  that  is,  it  increases. 

228.  A  quantity  which  is  fixed  in  any  given  problem  is  called 
a  Constant. 

Thus,  if  a  workman  receives  a  fixed  sum  per  day,  the  total  wages  due 
him  changes  from  day  to  day  if  he  works  and  remains  unpaid.  His  daily 
wage  is  a  constant ;  his  total  wages  is  a  variable. 

229.  A  change  or  Variation  in  one  quantity  usually  produces 
a  variation  in  one  or  more  other  quantities.  Such  variables 
are  called  Related  Variables.  For  each  value  of  one  variable 
there  is  a  corresponding  value  of  the  other  variable,  or  variables. 

Thus,  if  the  side  of  a  square  is  increased,  the  perimeter  and  the  area  of 
the  square  are  also  increased. 

230.  Variation  is  the  study  of  some  of  the  laws  connecting 
related  variables.  Instead  of  the  quantities  themselves,  their 
measures  in  terms  of  certain  units  of  measure  are  used. 

Thus,  distance  is  expressed  as  a  number  of  miles,  rods,  or  other  units 
of  lengtli  ;  weight  is  expressed  as  a  number  of  units  of  weight ;  area  is 
expressed  as  a  number  of  units  of  area. 

231.  One  quantity  varies  directly  as  another  when  the  ratio 
of  any  value  of  the  one  to  the  corresponding  value  of  the  other 
is  constant. 

Thus,  the  ratio  of  the  perimeter  of  a  square  to  the  side  of  the  square  is 
always  4,  becau.se  the  perimeter  is  4  times  the  length  of  the  side  ;  there- 
fore the  perimeter  varies  directly  as  the  side  of  the  square. 

232.  The  symbol,  (x,  is  read  "varies  as";  thus,  acchis  read 
"  a  varies  as  6." 


800  ALGEBRA 


If  xcjzy,  then  -  =  m,  where  m  is  a  constant,  expresses  the 

relation  between  any  two  corresponding  values  of  x  and  y. 
(See  §  231.) 

Since  -  =  m,  then  x  =  my. 

Either  equation  may  be  used  to  express  direct  variation. 


233.  One  quantity  is  said  to  vary  inversely  as  another  when 
the  product  of  any  value  of  the  one  and  the  corresponding 
value  of  the  other  is  constant. 

Thus,  the  time  and  rate  of  a  train  going  a  distance  d  are  connected  by 
the  equation  rt  =  d.  If  the  distance  remains  fixed,  then  the  time  varies 
inversely  as  the  rate  ;  for  example,  if  the  rate  is  doubled,  the  time  is  halved. 

If  X  varies  inversely  as  y,  then  xy  =  m,  where  m  is  a  con- 
stant, expresses  the  relation  between  them. 

If  xy  =  m,  then  also  a;  =  - .     Either  equation  may  be  used 

to  express  inverse  variation. 

234.  One  quantity  is  said  to  vary  jointly  as  two  others 
when  it  varies  directly  as  their  product.     If  x  varies  jointly 

X 

as  y  and  2;,  then  —  =  m,  where  m  is  a  constant,  expresses  the 
relation  between  the  variables. 

Thus,  the  wages  of  a  workman  varies  jointly  as  the  amount  he  receives 
per  day  and  the  number  of  days  he  works ;  for,  letting  W  equal  his  total 
wages,  w  his  daily  pay,  and  n  the  number  of  days  he  works,  then  W=  nw. 
Here  m  =  1. 

Again,  the  formula  for  the  area  of  a  triangle  is 
A  =  \ah. 

This  shows  that  the  area  of  a  triangle  varies  jointly  as  the  base  and 
altitude,     (Here  m  =  \.) 


RATIO,   PROrORTION,    AND   VARIATION  301 

235.  One  quantity  may  vary  directly  as  a  second  and  inversely 
as  a  third.     Let  x  vary  directly  as  y  and  inversely  as  z ;  then 

my 

x  =  -^ 
z 

expresses  the  relation  between  the  variables.  Notice  that  this 
combines  the  equation  for  direct  variation  of  y  and  inverse 
variation  of  z. 

236.  Variation  of  more  complicated  related  variables  needs 
to  be  expressed  sometimes. 

Example  1.     xcKy^  may  be  written  x  =  my^. 

Example  2.     af  oc  y^  may  be  written  ar'  =  7ny^. 

Example  3.  The  volume  of  a  circular  cylinder  varies  jointly 
as  the  altitude  and  as  the  square  of  the  radius.  This  may  be 
expressed  :  v  oc  ai-^,  or  v  =  ka'i'^. 

Example  4.     a  varies  directly  as  q,  and  inversely  as  (P. 

This  may  be  expressed :   a=  -—• 

EXERCISE  139 

Express  the  following  relations  both  by  means  of  the  symbol  oc 
and  by  an  equation: 

1.  The  area  of  a  rectangle  varies  jointly  as  the  base  and 
altitude.  ,        , 

2.  The  area  of  a  circle  varies  as  the  square  of  the  diameter. 

3.  The  volume  of  a  rectangular  prism  varies  jointly  a3  the 
length,  width,  and  height. 

4.  The  distance  a  body  falls  from  a  position  of  rest  varies 
as  the  square  of  the  number  of  seconds  in  which  it  falls. 

5.  The  interest  varies  jointly  as  the  principal,  the  rate,' and 
the  time. 


302  ALGEBRA 

Express  the  following  relations  by  mearis  of  equations : 

6.  The  rate  of  a  train  varies  inversely  as  the  time,  if  the 
distance  is  constant. 

7.  The  rate  of  gain  varies  inversely  as  the  capital  invested, 
if  the  total  gain  is  constant. 

8.  The  weight  of  an  object  above  the  surface  of  the  earth 
varies  inversely  as  the  square  of  the  distance  from  the  center 
of  the  earth. 

9.  The  per  capita  cost  of  instruction  for  pupils  in  a  school 
room  "Varies  directly  as  the  salary  of  the  teacher  and  inversely 
as  the  number  of  the  pupils. 

10.  The  volume  of  a  circular  cone  varies  jointly  as  the  alti- 
tude and  the  square  of  the  radius. 

11.  It  z  varies  jointly  as  x  and  y,  and  equals  f  when  .V  =  4- 
and  ic  =  f ,  find  z  when  y  =^  and  x  =  ^. 

Solution.     1.    According  to  the  conditions  z  =  mxy. 

2.  .  •.  -  =  m  •  -  .  ^ ,  or  m  =  - ,  since  s  =  ^,  when  x  =  |  and  x  =  L 

6  ^     5'  3'  "  ^  ^ 

3.  .  •,  0  =  I  xy,  substituting  |  for  m. 

4.  .  • .  ^  =  f  •  ^  ■  ^  =  J^,  when  x  =  |  and  y  =  l. 

Note.     In  such  problems,  first  find  the  constant,  as  in  step  2. 

12.  li  y  ccx  and  is  equal  to  40  when  x  =  5,  what  is  its  value 
when  x  =  9? 

13.  If  yocx^  and  is  equal  to  40  when  a;  =  4,  what  is  the 
equation  for  y  in  terms  of  a;? 

14.  If  X  varies  inversely  as  y  and  is  equal  to  |  when  y  =  f? 
what  is  the  value  of  y  when  a?  is  |  ? 

15.  If  (5  a;  -f  8)  oc  (6  2/  —  1)  and  x  =  6  when  y  =  —  3,  what  is 
the  value  of  x  when  y  =  7? 

16.  The  distance  fallen  by  a  body/  from  a  position  of  rest, 
varies  as  the  square  of  the  number  of  seconds  in  which  the 


RATIO,    PROPORTION,    AND   VARIATION  303 

body  falls.     If  it  falls  256  feet  in  4  seconds,  how  far  will  it 
fall  in  0  seconds  ? 

17.  The  interest  on  a  sum  of  money  varies  jointly  as  the 
rate  of  interest  and  the  principal.  If  the  interest  is  ??875 
when  the  rate  is  5%  and  the  principal  is  $3000,  what  is  the 
interest  when  the  rate  is  6%  and  the  principal  is  $2500? 

18.  The  principal  varies  directly  as  the  interest  and  inversely 
as  the  rate.  If  the  principal,  $4000,  produces  $250  interest 
at  4%,  what  princij)al  must  be  invested  for  the  same  time  to 
yield  $500  at  5%  ? 

19.  The  number  of  tiles  required  to  cover  a  given  area 
varies  inversely  as  the  length  and  width  of  the  tile.  If  it  ta,kes 
270  tiles  2  inches  by  5  inches  in  size  to  coyer  a  certain  area, 
how  many  tile  3  inches  by  6  inches  will  be  required  for  the 
same  area  ? 

20.  The  number  of  posts  required  for  a  fence  varies  inversely 
as  the  distance  between  them.  If  it  takes  80  posts  when 
they  are  placed  12  feet  apart,  how  many  will  be  required 
when  they  ate  placed  15  feet  apart  ? 


XVIII.    GENERAL   POWERS   AND   ROOTS 

237.  In  the  preceding  chapters,  only  positive  integers  have 
been  used  as  exponents.  The  fundamental  definition,  when 
w  is  a  positive  integer^  is : 

a^  =  a  '  a  '  a  '"  a     (m  factors).  (§  15) 

More  general  powers  occur  in  mathematics. 
When  m  and  n  are  positive  integers : 
I.     Multiplication  Law.        Just  as    a^^  •  a*  =  a^*,  (§  53) 

so  a'"  X  a"  =  a'"+". 
II.     Division  Law.  Just  as  aF  -i- o^  =  a^,  (§  68) 

soa'"  -^(f=:if"-". 
-  III.     Power  of  a  Power.  Just  as       (a^  =  a'%         (§  89) 

so     {(f"y  =  <f", 

IV.     Power  of  a  Product.         Just  as       (abf  =  a%%        (§  89) 

so      {ahy  =  a"l/'. 

V.     Power  of  a  Quotient.       Just  as       (^-Y  =  ^,  (§  89) 


so 


3"  _cP 
~6^ 


EXERCISE  140 

•  In  the  following  list  of  examples,  the  literal  exponents 
denote  positive  integers.  Find  the  results  of  the  indicated 
operations  by  using  the  five  laws  above: 

1.    Q^^ '  x^.  4.    m^"  •  m".  7.    c"~'*  •  c". 

3.    2/*  •  2/".  6.    6''+^ .  61 

304 


GENERAL   POWERS   AND   ROOTS  805 


10. 

r-* .  r+». 

19. 

c'-^H-c^. 

28. 

(-<»»^)*. 

11. 

«r+" .  w"*-". 

20. 

d'r+S^flr^ 

29. 

{i/z'w)\ 

12. 

g^   r^l,  gr. 

21. 

zr+*^z'+\ 

30. 

{mhiyy. 

13. 

x^'-i-x'\ 

22. 

^«+6  _^  ^«-i 

31. 

(a'y. 

14. 

x^'^af. 

23. 

^,j"»+»  ^  M,"-«. 

32. 

(b-y. 

15. 

f^y\ 

24. 

^2«-r+l_^^r^ 

33. 

(-  c^drf. 

16. 

m=^  -f-  m". 

25. 

(a.«)^ 

34. 

wr- 

17. 

a'*"  H-  a\ 

26. 

(yj' 

35. 

{r*6^y. 

18. 

If^^-hb*. 

27. 

(my. 

36. 

(x-y-y. 

37. 

'©■■ 

40. 

43. 

©•• 

38. 

if)-- 

41. 

(-?)■• 

44. 

(?)■■ 

39. 

©'■ 

42. 

m- 

45. 

238.   Only  square  and  cube  root  have  been  used  in  the  pre- 
ceding chapters.     More  general  roots  occur  in  mathematics. 

Just  as  V^  indicates  the  cube  root  of  «,  (§  91) 

so  -y/x  indicates  the  ?ith  root  of  x. 
n  is  called  the  Index  of  the  rooti 

The  nth  voot  of  x  is  an  expression  whose  nth  power  equals 
x ;  that  is,  {Vxy  =  x. 

Thus,  -y/^^  =  x^,  since  (a^y  =  x'^ 

V^  =  x\  since  (a;*)*  =  3^. 
■^-a^iyi  =  _  :i^f^  since  (-  o^^/O^  =  -  ai^y*. 

Notice  that  the    exponents   of   the   monomial    und^r  ^thie 
radical  sign  are  divided  by  the  index  of  the  roiot. 


80a  ALGEBRA 

Historical  Note.  A  symbol  for  extracting  a  root  did  not  apjiear 
until  the  fifteenth  century.  In  Italian  inatheniatics,  the  fii-st  letter  of  the 
word  Radix,  meaning  the  root,  was  used  to  indicate  the  square  root :  thus, 
R.  Presently  there  were  used  R.2«,  Ii.3°,  etc.  to  indicate  the  square, 
cube,  and  other  roots.  Chuqwet,  a  French  mathematician  of  about  1500, 
used  R-^,  R^  etc. 

In  Germany,  a  point  was  placed  before  a  number  to  indicate  that  its 
square  root  was  to  be  taken.  Two  points  were  used  to  indicate  the  fourth 
root,  and  three  the' third  root.  Reise,  1492-1559,  replaced  the  point  by 
the  symbol,  y/,  to  indicate  the  square  root,  and  Rudolph,  1625,  used  the 
symbol,  ys/i  for  the  fourth  root.  Stevin,  1548-1620,  used  the  better 
symbols  :  VCD.  ^(S),  etc.  Girard,  1590-1632,  used  :  ^/^  ^,  etc.  Des- 
cartes used  the  vinculum  to  indicate  what  numbers  were  a^ected  by  the 
root.  A 

EXERCISE   141 


Dei 
1. 

;ermine: 

10. 

11. 

12. 
13. 
14. 
15. 

16. 

17. 
18. 

V4  x'. 

19. 
20. 
21. 
22. 
23. 
24. 

25. 
26. 

27. 

-^o^'. 

2. 

^Ux\ 

^32, 

a/6^'cI 

3. 

^_^nyl0.n 

4. 

</a}n''. 

--'■^'5. 

-v/64  a%\ 

Va;H'\ 

6. 

{■■■  ■  ■ 

7. 

^/625  a%\ 

»/243 

Vx'^y^. 

V7- 

'-8. 
9. 

239.  Negative  numbers,  fractions,  and  zero  are  also  used  as 
exponents.  So  far,  such  expressions  as  a"^  and  a^  have  no 
-Hieaiiing,  as  it  is  impossible  to  write  a  as  a  factor  —3  times 
or   five  thirds  times.      It  is  desirable  that  such   expressions 


GENERAL   POWERS  AND   ROOTS  897 

should  obey  the  laws  of  exponents  considered  in  the  preceding 
paragraphs. 

.  (a)  Meaning  of  a  Fractional  Exponent.      If  a^  is  to  obey  the 
multiplication  law,  (§  227),  then 

at .  a'3  .  at  =  fti  +  I  +  5=:aV==c[65 

that  is,  a'  cubed  gives  a^,  or  a^  is  the  crtbe  root  of  a^. 

This  suggests  the  definition  :  in  a  fractional  exponent,  the 
denominator  denotes  a  root  and  the  numerator  d.pnotes  a  power.  . 

Thus  :  X*  =  V.'B^ ;  y"  =  V/ ;  Zn=  -\/z"*. 

% 

(b)  Meaning  of  a  Zero  Exponent.     If  a"  is  to  obey  the  same 

law,  then  a'^X  a^=  a'""'*^=  a*";  or  a"=  a'"H-  a'"=  1. 

This  suggests  the  definition:  the  zero  power  of  any  number, 
except  zero,  isl. 

Thus:  5«=1;  a/»  =  l;  (  -  65)«  =  1. 

(c)  Meaning  of  a  Negative  Exponent.  If  a~^  is  to  obey  the 
same  law,  then  a~*  •  a?  =  a"^'^^  =  a^  =  1:  or  a~^  =  — . 

This  suggests  the  definition:  a""=  --■.   ■ 

a"" 

Thus  ,  ,      .       ,  ^ 

Historical  Note.  In  the  note  following  §  14,  credit  is  g^ivpn  to 
Herigone  for  having  grasped  the  idea  of  an  exponent  and  for  introducing 
a  rather  good  notation.  As  early  as  1484,  another  French  mathematician, 
Chuquet,  had  had  some  idea  of  an  exponent'and  had  written  expressions 
involving  a  form  of  negative  exponent  and.  also  the  zero  exponent.  His 
ideas,  however,  did  not  spread  far.  Other  attempts  to  introduce  general 
exponents  were  made  between  that  time  and  the  time  of  Newton.  .  T^o 
Newton  must  be  given  the  credit  for  having  finally  fixed  the  present  form 
of  writing  the  various  kinds  of  exponents.     --  .  i 


808  ALGEBRA 

EXERCISE  142 

Express  with  radical  signs : 

1.  a^.       3.   Ax^.        5.    a^-b^.        7.    6  x^yK        9.    ab^c^^dh 

2.  bK       4.   9a6i     6.    mM.        8.    8a^mi       10.    Sa;"^^^''^. 

Express  with,  fractional  exponents : 


U.  ^a^ 

13.    Vm^. 

15.    2^w«. 

17.    ^o^.  ^6^. 

12.    a/^ 

14.    ^5"». 

16.   4-</^«. 

18.    -y/m^ .  -v^. 

Write  with  positive  exj)onents  and  radicals 

19.  a'b-'. 

20.  m-V. 

21.  aV«. 

24.  a-«6-V. 

25.  a~fei 

26.  771^ '  ?ri 

2"^ 

x-^ 
nvx 

22.  2a-'x-\ 

23.  4a-«6'-i. 

27.  a;i-y-«. 

28.  z^'^^x-^. 

Write  with 

positive  exponents  and  radicals 

and  simplify : 

35.  4i 

40.  8-i 

' ..  r:. 

50.  8i 

36.  9i 

41.  36-*. 

51.  (-  64)1 

37.  27*. 

42.  (-125)- 

52.  144-*  .  24. 

38.64* 

43.  32-*. 

3-2 
48.  16  .  2-\ 

53.  39  .  169-*. 

39.  (-  8)1 

44.  8ri 

49.  2=^  •  2-^ 

54.  1000.100'^ 

Perform  the  following  indicated  operations  : 
55'.  a^ '  a-^  .  57.  a;"*  •  y-^.  59.  m*  •  m^. 

66.  a» .  a-K  58.  2/-V  •  y^i^"^,  «0.  W^  -Lmi 


GENERAL   POWERS   AND   ROOTS  309 


61. 

nt.ni. 

69.  z^^zK 

77. 

(m-%-«)l 

62. 

2  n-i  .  n-\ 

70.   ^o^  -i-w. 

78. 

(rV)-^. 

63. 

x^  .  xK 

71.   (a:^)«. 

79. 

(i>V)i 

64. 

a  '  Va. 

72.    (7/^2'^)l 

80. 

{mhi')K 

65. 

.T-"  .  -^. 

73.  (m~^  .  n^)^. 

81. 

(x-WK 

66. 

m-  -  Vm. 

74.  (x-^y. 

82. 

(«-W)+l 

67. 

x-^-i-x-^. 

75.  (a-^6^)«. 

83. 

-v^m  •  Vn. 

68. 

z-*^z-\ 

76.   (a2&4)i. 

84. 

v^.^. 

85. 

(J 

+  ah^  -\-b^)(J- 

•b^). 

86. 

(4.x-^-6x-'-\-9)(2 

x-^  +  S)^ 

87.  (2  a-i  -  7  -  3  a)(4  a-^  +  5). 

88.  (a;-^  +  2)(x-^  -  2).  92.  (05-  -  r)(^"  +  r)- 

89.  {x^  4-  7)(a;^  +  3).  93.  ^r  -  2/")^- 

90.  (a«  +  ^'O  -^  (^^^  +  '->^)-  94.  (a;2-  -f:  y^)2. 

91.  (a-*  —  1)  -^  (ot"^  —  1).  95.  (a^  +  4)(aj'»  —  7). 

96.  (x""  +  3)(a;'«  —  5). 

97.  (aj^*"  -I-  2  a;"'  4-  l)(a;'^  +  1). 

98.  (a^"  +  a"6"  +  62'»)(a-"  —  a'^^"  +  ^'"). 

99.  (ar^  +  3  a:^"^  4-  3  aj^  4-'  /)  -^  (a?"  +  Z/)- 
100.  (r^  — s^*)-H(r"*  — s*). 


MISCELLANEOUS   EXERCISES 

A.     ADDITION  AND  SUBTRACTION 

Simplify : 

1 .; .  ax  4-  hx  +  ex.  3.    2  aa^  +  6?/^  -f  3  dx^  +  c^/^ 

2.   mx  +  Zi!/  —  nx  +  A;i/.  4.  px^  -f  ^i/^  +  »'^^  +  5  2/^. 

6.  (a  -f  6)  a;  H-  (2  a  —  Z;)  cc. 

7.  3  (m-n)^-4(m-w)H  5(m  +  ?i)-  7(m+w)+  8(m-?i)'l 

8.  Ada  3(r  +  s)+  6(i  +  s)-  7  (?•  +  0^  ^0'  +  ^•)— 3(t  -f-  s)  + 
5(r  +  0,  and  -5(r  +  6')  -  2(i +- s) +4(r  +  ^). 

9.  Add  7  a^«  -^ h  x^'  +  S-oj"  -  1,    -  2if««  +  5  -  3  a.-  +  Grc^n^ 
and  —  5  ic^"  —  4  a;^  +  8  X""  —  7. 

10.  From   7  a;"  —  3  a^"-^>  +  5  a;^"-^)  +  4   subtract    2  a;"  +  5  -|- 

11.  By  how  much  does  as'*  4-3  aj^"-^^?/  +  4  ir^"-^y  +  5  x^^'-^hf 
exceed  «"  —  2.a;^"-^^2/  +  5  a?^"-^)?/^  -|-  8  a;("-^>2/l 

12.  Subtract  ,  5  (a;  +  2/)^  +  6(a;  +  2/)  4-  9   from   13  (x  4-  2/)^  4- 
2(a;4-2/)-4: 

13.  Subtract  —  xT -\-  nx^^'-'^^y  4-  (n  —  1)  x^^^-^^y-  from  0. 

14.  How  much  must  be  added  to  x'^  —  x^y  4-  x-y^  4-  xy^  4-  y*  to 
give:  (a)  x*—y*?     (b)  x*-\-y*? 

15.  Subtr act  aar  4-  2  fta-?/  +  C2/^  4-  cZa?  4-  62/  4-  /  from  Ax^  4-  2  ^a-?/  4- 
Cy^  +  Dx  +  Eyi-F. 

810 


MISCELLANEOUS   EXERCISES  311 

B.     EVALUATION  OF  ALGEBRAIC  EXPRESSIONS 

Let  a  =  4,  6  =  5,  c  =  6,  p  =  2,  m  =  2,  n  =  3 ;    iiud  the  nu- 
merical value  of  the  following  : 

1.    a".  2.    h"".  3.    c''->rp'^.  4.    6'*  — 4/>'". 

'263*  *   4  a- 36 -f  6  c*  '   \^a  -  6y  * 

8.    h^-\-c^-2hp.  9.    c(a+6)«.  10.    2a  +  (yi-l)j9. 


When  ^  =  28,  r  =  6,  s  =  4,  and  tt  =  3|: 

11.  Find  F:  (a)  if  F=7r?-7i;    (6)  if  V^^irr". 

12.  Find  S:  (a)  if  aS  =  2  7rr/i ;  (6)  if  /S  =  2  tti^  +  2  ttt^. 

13.  Find  Fif  F=  ^  7r/i(r2  +  s^  +  rs). 

14.  Find  a;  if  w  =  1000,   T  =  35,  «  =  15,  and  s  =  50,  when 

s 

15.  Find  F  if  P  =  500,  i>  =  5,  d  =  i  and  ^  =  72,000,  when 

Ed' 

C.    PARENTHESES 
Remove  the  parentheses  and  combine  terms : 


1.  c-[2  c-(6  a- 6)- ;c- 5  a-f-2  6 -(-5a  +  6a- 36) j]. 

2.  a  -  (2  a  -  [3  a  -  54  a  -  5  a  -  If]). 

3.  X  -[y  -  Ix  -  z  -  X  -  y  +  zl  +  (2  X  -  {-  X  +  yl);i. 

4.  5a;-[2aj-(-a;- j2a;-^~^S-3a;)-3a;]. 

5.  a  —  I  —  a  ~  [—  a  —  (  —  a  —  I  —  a  —  a  —  l\)']l. 

6.  28  -  5  -  16  -  (-  4  +  [55  -  31 +47])  J. 

7.  a:  -  (2/  -h  2r  -  [a:  -  (_  a;  -  2/)  +  2])  -h  fz  -  2  X  -  2/|. 

8.  2n-[3?i-|4n-?r^^i -(-5n~9)]. 


812  ALGEBRA 


D.     MULTIPLICATION 


Perform  the  following  indicated  operations  : 

1.  (x-y  +  zf-(x  +  y-  z)\  3.    {a  -  h  -  c -{- d)\ 

2.  {2x-\-  3)2(2  X  -  3)2.  4,  («  +  hf  -{a-  hf. 

5-  (« -y){y-^)-{^- ^) (y-^)-(x- y) (x - z). 

6.  {^x-^yf-n{x-y){x-5y), 

7.  f  («  -  &)  («'  +  a'6  +  «&'  +  &') H K  +  ?>')'-  2  a^ft^j. 

8.  {a-\-h+cy-{a  +  hf-G(2a  +  2h  +  c). 

9.  2(a  +  2  a?) (a  -  2  x)  -[(a  +  2  a;)^  +  (a  -  2  it')^]. 

10.  {a-{-x){o?-^)[_a^-x{a-x)']. 

11.  (aj»-f6)^.  13.    {x^  +  y^f. 

12.  (a**  —  6")  (a"  +  ft**).  14.    (ar'^«  +  a^«  +  a;")  .  (a;»  +  1). 

15.  («2'*  +  3 !»-«  4-  3  a;"  4-  1)  •  (•'«"  +  1). 

16.  lut  +  i/^2-,  _  [-^(^  _  1)  +  |y.(^  _  i)2-j. 

E.    DIVISION 
Divide : 

1.  70a-50-a^-37  a2by6a-5-a3-2a2. 

2.  a^-a^'^-a^/^  +  ^^^by  a2-2a6  +  62. 

3.  a^  H-  2/^  +  2;^  —  3  xyz  hy  x  -[-  y  -\-  z. 

4.  a3-86-^-27c^-18a6cby  a-25-3c. 

5.  8m^-14m2-18m  +  21  by  4m^  +  6m-7. 

6.  (2m2-m-l)(3m2  +  m-2)  by  (2  m  +  1)(3  m- 2). 

7.  I  a^-l  a^b  +  ^ab'-ib^hy  ^a-ib. 

8.  (x-yy-{-lhy(x-y)-\-l. 

9.  6(m  +  7i)2-(m  +  n)-15by  3(m  +  w)-5. 

10.  ix'-lx^  +  ^x-^hj^x^-^-^x-^. 

11.  (ar^'"  +  i»2m  _|.  ^>j  _^  ^«  ;^2.    (a^4.5a;«4-6)-^(af-^-2). 

13.  (f-^Sy^--^Sy--^l)-h(y^'{-l). 

14.  (a:^"*  —  y^)  ~r  {xT  —  y™).         15.    a;*"  —  -^^  (pf*  -|-  ^'''). 


MISCELLANEOUS  EXERCISES 


313 


F.    FRACTIONS 

Simplify  the  following  by  performing  the  indicated  operar 
tions : 

1.   12<x  — 5  — ^^ -^  , 

3a  +  2 


2. 


+  R 


3       fr'  +  ^'T    _(a2_^a6  +  6'^- 


1 


3r 


6rs 


7. 


9. 


(3r-25)      (3r-2s)2      (3r-2s)3 


y    ^ 


8-    IF+S.^VT^l^-F. 


11.   «ii^_|_c4-f^ 


1-(H 

2(ac-M) 


10    (a^  +  2y)'^-6a;y 


+ 


12. 


a  —  6     c  —  d      (6  —  a)  (c  —  d) 
3a     _  4a 


a2_3a4-2  "  a=^-7aH-10     a^-Ga-f-S 

13.    f2a  +  3— ?i^Vf2a-3  +  5i2«-3}\ 
V  2aH-3>'      V  ^    2a^6  ; 

3a;  +  2 


14. 


5x-l 


6x2_3,_i2     10x2_i9^,^g 

m^  +  4  mn  -|-  4  n^  *  \m^  —  4  w^  *  rn^  +  mn  —  Qn^j 

16    ar^-2a?.V  +  4y^4a:^-9/  .  2a;-3y 
2a;  +  32/        '    x'^^f    ''a^-4:y^' 


17. 


X—  m      X  ■\- 


m.  —  n" 


a;  4-  n      a;  —  m      (oj  —  m)  (a;  -f  n) 


314 

18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 


1/1^1 


a\x  —  a     x-\-2aJ     x^-\-ax—2d 


ALGEBRA 
3 


4m-3 


+ 


2m-5 


6m2  +  13m-5      12 m^  +  Sm- 3 

2 


x2-3 


' ^-\^(. I 1_>) 

x  +  2     x-lj      \x'-3x-\-2     x-2) 


4- 


(a  —  6)  (a  —  c)      (6  —  c)  (6  —  a) 

?— 1 + 1 

(x-y)(y-^)    (y-z)(^-^)    (z-^)(^-y) 
g  +  c (g  +  &)       .  (&4-C) 

(a-5)(6-c)  ,  (6-c)(c-a)      (c-a)(a-6)' 


1 


1 


a(a  —  b)(a  —  c)      b(b—c)(b  —  a)      c(c  —  a)(c  —  b) 

1 1  2ft 

(a  -  &)      a  +  b     (5^  -  ft^  * 

a-1      g  +  lj    *   |a-l"'"a4-lj    '   J(a  +  l)2_2J 


(l ^^— Vl- 

\       1*^  —  rs  4-  sy  \ 


x'^-2  jfy^  4-  y^  ^  (x^  -  ;V^  ^  (a^  +  y)^  -  2  a;y' 


a;?/ 


«y 


1  +^+  2 


1-x     l-\-x      1  +  a^ 


First,  add  the  first  two  frac- 
tions, then  add  to  the  result  the 
third  fraction. 


1   +  ^ 


2  a  ^aW 


a-b     a-\-b     g2-f  62     a' +  b' 
1  1     +     1  1 


-•  Combine    the    first    two 

, '  .       fractions ;     then     the    last 

t-1      t  +  1      t-2      t-\-Z  ^^^,^^  ^^^  ^^^  ^jjggg  results. 

_1 1  d  d 

c-d      c  +  d      c2-fd2      c^_(p' 


MISCELLANEOUS   EXERCISES  315 


33.  _!_  +  -J_  +  -2y_  +  _4/_. 
y  —  x     y-\-x     f-^^     .V*H-a;* 

34.  -i L_+_25__„4a^. 

a-h      a  +  ^>     a'  +  ft'     «*  +  ^' 

112  2 


^-1      «+l      «+2     i-2 
G.   FRACTIONAL  EQUATIONS 

7a:^  +  10a;-24  5      ^„ 

{x^\f  (x'  +  l)        ■ 

6  ^       .2(2a!  +  l) 

4     (a?H-l)(a;-h3)^a;-6  ^     3a?^  f  5a;-4^3a;  +  5 

(a;-f5)(a;  +  7)      aV+2'  '    4a;2_3^_|.2     4a;-3 

^     a;  —  2      «  —  3      a?  —  5      u?  —  6 


«  — 3      a;  — 4      a;  — 6      x  — 7 

First  combine  the  fractions  in  the  first  member  ;  then  the  fractions  in 
the  second  member. 

a;  — 4     x  —  h     x  —  \      x  —  2 


x—5     x—6      x—2      x—S 

-     m-\-2      m  +  6  _  m-\-5     m  +  3 
'    m  +  3      m-\-l      ?/i  -f  6      m  +  4 

Make  this  example  like  Example  6  by  transposing. 

1  a  3a-4 


9. 


a  +  1     a^-a  +  l      a^  +  l 

r  +  2  2(r-l)  ^ 

r2^-4r  +  3      7^-\-r-Q     r* 


316 


ALGEBRA 


H.   LITERAL  LINEAR  EQUATIONS 

2.   2(x-b)(2a-Sb-Sx)-(2a-Sx)(b  +  2x). 

3.  (x  +  ay-^(b-j-cy=(x-ay+{b-cy. 


0. 


5. 


{x-2a-^3by 

ax  bx    _ 

x-{-b     x  +  a 

2(a  +  b)  ^x  +  b 

X  x  —  b 

1 


-(x-2a)(x-\-Sb)-6ab  =  0. 
a  +  b. 


7. 


b      b_ 


C        C— ft_A 


x—c      x  —  a     a—x 


a 


9. 

10. 

11. 
12. 


+ 


x-\-a 
1 


8     (2^-3m)^^a;-39?i 
(2x-3ny       x-Sn' 


2x—a—b 


(x  —  a)      (x  —  b)      x(x  —  a  —  b) 


x  +  Aa  +  b     4:X-\-a-\-2b 
x-\-  a-\-b         x-\-a—  b 


5. 


T-\-agp(h-cQ) 

Nn         r 


Mm     p  —  r 


agph. 
Find  r. 


Find  r. 


1. 


I.   SIMULTANEOUS  LINEAR  EQUATIONS 


±i+s=2x-y^^. 

7  4 


Sx- 


x-hy 


2y-S 


=  2y-4.. 


^^-^^  +  3  =  0. 


7 


3. 


4. 


1. 


S-^_2x±3^y±S 
5  4  4 

-5 


11  3 

5  m  +  6     11 7* 


10 
7  m     55  r 
~5" 


21 
12 


11. 


25 


37. 


6. 


2(3a  +  6)_i(2a  +  6  +  l)=iV- 
3a-f(4a  +  &  +  |)  =  56. 

f3s-i(2s  +  «  +  6)  =  5^. 


20-i(2i-s)  =  -8. 


MISCELLANEOUS   EXERCISES 


317 


8. 


9. 


10. 


X  4-  ay 
6-2 


=  a. 


hx  -\-  ay 
h 

a. 

-^-^  =  -1. 

m      n 

X       y  _. 

3m     6n 

2 

3* 

3a^36 

+  b. 

x-y  =  2{a^ 

-b^ 

2x-b     3x 

-y. 

a 
2x-b 


a  +  26 
a-2y 


13. 


y    * 

22     2/ 


14. 


x     y 


1+1=1, 

y     z      a 


z      X 


15. 


a:-|-?/  +  «  =  —  8. 

y-^z-\-u  =  h. 

^Z-\-ll,-\-X  —  —  10. 


16. 


2*_2a;  =  -13. 
i»-3?/  =  13. 
2/  —  4  z  =  5. 
z-5w=23. 


a  6 

12a;-4?/  +  2;  =  3. 

11.  {x  —  y  —  2z=^  —  l. 
5x-2y  =  0. 
3x  —  y  —  z  =  7. 

12.  <x  —  Sy  —  z  =  21. 
x-y-Sz  =  27. 

J.    CLOCK  PROBLEMS 

Around  the  clock  face  there  are  12 
hour  spaces  and  GO  minute  spaces. 
While  the  minute  hand  passes  over 
the  60  minute  spaces,  the  hour  hand 
j)asses  over  1  hour  space  or  5  minute 
spaces.  Therefore,  in  any  given  time, 
the  hour  hand  of  a  clock  moves  one 
twelfth  as  far  as  the  minute  hand. 

Thus,  while  the  minute  hand  moves  over  24  minute  spaces,  the  hour 
band  moves  over  2  minute  spaces. 


318 


ALGEBRA 


When  the  minute  hand  makes  the  complete  revolution  of  60 
minute  spaces,  it  passes  through  an  angle  of  360° ;  therefore 
an  angle  of  6°  corresponds  to  each  minute  space. 

For  example,  in  15  minutes,  the  minute  hand  moves  through 
15  X  6°,  or  90° ;  also,  at  3  o'clock,  there  is  an  angle  of  90°  be- 
tween the  hands,  since  they  are  separated  by  15  minute  spaces. 

1.  Tell  how  many  minute  spaces  the  hour  hand  will  move 
while  the  minute  hand  moves  : 

(a)  36  spaces ;  (b)  48  spaces ;  (c)  x  spaces. 

2.  Tell  how  many  minute  spaces  the  minute  hand  will  move 
while  the  hour  hand  moves : 

(a)  2  spaces ;  (b)  3^  spaces ;  (c)  x  spaces. 

3.  What  angle  do  the  hands  of  a  clock  make : 

(a)  at  2  o'clock  ?  (b)  at  4  o'clock  ?  (c)  at  6  o'clock  ? 

4.  At  what  time  will  the  hands  of  a  clock  be  together  be- 
tween 7  and  8  o'clock  ? 

Solution  :  1.  At  7  o'clock,  the  hour  and 
minute  hands  are  separated  by  35  minute 
spaces.     (See  the  light  lines.) 

2.  When  they  are  together,  the  minute 
hand  must  have  moved  over  these  35  minute 
spaces  and,  besides,  over  the  spaces  through 
which  the  hour  hand  has  moved.  (See  the 
heavy  lines  of  the  figure.) 
3.    Let  X  =  the  number  of  minute  spaces  moved  by  the  minute  hand. 

.'.  —  =  the  number  of  minute  spaces  moved  by  the  hour  hand  in 
12 

the  same  time. 

.'.    X  =1  36  +  — .     (See  figure  and  step  2.) 
Complete  the  solution. 

5.  At  what  time  between  6  and  7  o'clock  will  the  hands  of  a 
clock  form  an  angle  of  60°  for  the  first  time  ? 

Solution  :  1.    At  6  o'olock  the   hands  are  separated  by  30  minute 
(See  the  light  lines  in  the  figure. ) 


MISCELLANEOUS  EXERCISES 


319 


2.  When  they  are  separated  by  60"",  they  will  be  separated  by  10  min- 
ute spaces,  since  6°  correspond  to  1  minute  space.  (See  the  heavy  lines 
of  the  figure. ) 

3.  Let  n  =  the  number  of  minute  spaces  moved  by  the  minute  hand. 

.  •.  —  =  the  number  moved  by  the  hour 
12 

hand  in  the  same  time. 

.-.  The  number  of  minute  spaces  from  12 

around  to  the  final  position  of  the  hour  hand 


and  also 

.-.30  +  — 
12 


30  +  -^; 
12' 

n  +  10. 
n  4- 10. 


6.  At  what  time  between  9  and  10  are  the  hands  of  a  clock 
together  ? 

7.  At  what  time  between  4  and  6  are  the  hands  of  a  clock 
together  ? 

8.  At  what  time  between  1  and  2  are  the  hands  of  a  clock 
opposite  each  other  ? 

9.  At  what  time  between  5  and  6  are  the  hands  of  a  clock 
separated  by  an  angle  of  90°  for  the  first  time  ? 

10.   At  what  time  between  8  and  9  o'clock  are  the  hands 
separated  by  an  angle  of  120°  ? 


K.     MIXTURES 

When  drugs  come  from  the  wholesalers,  their  strength  is 
known.  Thus,  alcohol  is  95  %  pure ;  this  means  that  one  gal- 
lon of  it  contains  ^^  gallon  of  pure  alcohol  and  j^  gallon  of 
water.  When  used  by  the  druggist,  drugs  are  sometimes  di- 
luted by  adding  water  or  other  ingredients. 

Example  1.  How  much  water  must  be  added  to  1  quart  of 
some  75  %  alcohol  to  reduce  it  to  40  %  alcohol  ? 


320  ALGEBRA 

Solution  :  1.  In  the  1  quart  of  76%  alcohol,  there  is  .75  quart  of  pure 
alcohol.  .• 

2.   Let  w  =  the  number  of  quarts  of  water  to  be  added. 

3.-  . *.  (-MJ  +  1)  =  the  total  number  of  quarts  in  the  new  mixture. 

4.  40 7o  of  the  new  mixture  is  to  be  alcohol ;  i.e. 
.40(10  +  1)  is  alcohol. 

.-.  .40(10  i}-l)=. 76. 
i^.'b.   Solving  this  equation, 
ry      ■  -MJ^f^orl. 

V  •  Ze.'l  quart  of  water  must  be  added. 

Check  •  1.   The  total  mixture  is  then  1|  or  ^^-  quarts. 

2.  There  was  .75  or  f  quart  of  alcohol. 

3.  Is  f  quart  40%  of  y-  quarts  ? 

Example  2.  How  much  of  95  %  alcohol  and  of  pure  water 
must  be  taken  to  make  a  mixture  of  two  qwarts  which  shall 
contain  45  %  of  alcohol  ? 

Solution  :   1.  Let  w  =  the  number  of  quarts  of  water. 

2.  .  •.  (2  —  w)=  the  number  of  quarts  of  95  %  alcohol. 
.'.  .95(2  —  io)  =  the  number  of  quarts  of  pure  alcohol. 

3.  But  45  %  of  2  quarts  is  to  be  the  amount  of  pure  alcohol. 

.-.  .95(2-10)  =.45  X  2. 
.-.  1.90- .95 10  =  .90. 
.-.  -.95  10  =-1.00. 
.-.  w=  l^  =  the  amount  of  water, 
and  (2  —  to)  =  ^f  =  the  amount  of  95  %  alcohol. 

Check  :  Is  95  %  of  if  the  same  as  45  %  of  2  ? 
'        '  *xH  =  t¥^;  AVx2  =  i^^. 

EXERCISE 

1.  How  much  water  must  be  added  to  one  quart  of  95  %  al- 
cohol to  make  a  mixture  which  shall  be  50%  alcohol? 

2.^^ Tincture  of  arnica  of  standard  strength  is  20%  pure; 
this  means  that  it  contains  20  %  of  arnica  and  80  %  of  alcohol. 


MISCELLANEOUS  EX^ERCISES  321 

How  much  alcohol  must  be  added  to  one  quart  of  the  20  % 
mixture  to  make  a  15  %  mixture  ? 

3.  How  much  alcohol  must  be  added  to  a  pint  of  belladonna, 
10  %  pure,  to  make  a  mixture  8  %  pure  ? 

4.  Stronger  rose  water  is  25  %  pure.  How  much  water 
must  be  added  to  2  quarts  of  stronger  rose  water  to  make  a 
10%  mixture? 

5.  Hydrochloric  acid  may  be  bought  chemically  pure.  How 
much  water  must  be  added  to  one  half  gallon  of  chemically 
pure  acid  to  make  a  10  %  mixture  ? 


INDEX 


A,  the  symbol,  48. 

Abscissa,  212. 

Absolute  value,  26. 

Algebraic  expression,  17  ;  value  of 

an,  17. 
Angle,  15  ;  right,  15  ;  straight,  15. 
Angles,  complementary,  15  ;  sum  of 

in  a  triangle,  17 ;  supplementary, 

16. 
Ascending  powers,  39. 
Axis,  horizontal,  211  ;  vertical,  211. 

Base,  18. 

Binomial,  38  ;  square  of  a,  117. 

Braces,  55. 

Brackets,  55. 

Cancellation,  in  an  equation,  98  ;  in 

a  fraction,  161. 
Changing  signs,  in  an  equation,  99  ; 

in  a  fraction,  163. 
Clearing  of  fractions,  185. 
Coefficient,  34  ;  numerical,  34. 
Complement  of  an  angle,  15. 
Complex  number,  280. 
Conditional  equation,  96. 
Coordinates,  212. 
Cube,  of  a  monomial,  HI ;  perfect, 

112. 

D,  the  symbol,  48. 

Degree,  of  a  monomial,  154  ;  of  an 

angle,  15  ;    of  an  equation,  221  ; 

of  a  polynomial,  164. 
Descending  powers,  39. 


Elimination,  by  addition  or  subtrac- 
tion, 223  ;  by  substitution,  225. 

Equation,  7,  96 ;  canceling  terms 
in  an,  98  ;  changing  signs  in  an, 
99 ;  complete  quadratic,  258 ; 
conditional,  96  ;  degree  of,  221  ; 
fractional,  185 ;  graphical  solu- 
tion of  an,  259 ;  identical,  96 ; 
indeterminate,  222  ;  integral,  185  ; 
linear,  221  ;  literal,  200  ;  mem- 
bers of  an,  7  ;  of  first  degree,  97  ; 
properties  of  an,  97  ;  pure  quad- 
ratic, 254  ;  quadratic,  254 ;  simple, 
97  ;  solving  an,  7  ;  transposition 
in  an,  98. 

Equations,  dependent,  222  ;    incon- 
sistent, 222 ;    independent,    222 
simultaneous,  222. 

Exponent,     18 ;      fractional,     307 
negative,  307  ;  zero,  307. 

Exponents,  law  of  division  of,  87 
law  of  multiplication  of,  66. 

Expression,  algebraic,  17. 

Extremes,  291. 

Factor,    3 ;    common,    11  ;    highest 

qommon,  155 ;  to,  110. 
Factors,  prime,  110. 
Formula,  19. 
Formulae,  deriving,  202. 
Fourth  proportional,  292. 
Fractions,    160 ;    clearing  of,    185 ; 

equivalent,  167. 
Fulcrum,  193. 
Fundamental  operations,  18, 


323 


324 


INDEX 


Graph  of  equation  with  two  varia- 
bles, 216. 

Graphical  representation,  206. 

Graphical  solution  of  equations  with 
one  variable,  259. 

Graphs,  206. 

Grouping,  symbols  of,  55. 

Horizontal  axis,  211. 

Identity,  96. 

Imaginary  number,  278. 

Imaginary  numbers,  addition  and 
subtraction  of,  280. 

Imaginary  roots  of  a  quadratic  equa- 
tion, 278  ;  meaning  of  on  graph, 
281. 

Imaginary  unit,  279. 

Inconsistent  equations,  220,  222. 

Independent  equations,  218,  222. 

Indeterminate  equations,  216, 
222. 

Index,  113,  305. 

Integral  equations,  185. 

Left  member  of  an  equation,  7. 

Lever,  193. 

Like  terms,  35. 

Linear  equation,  221. 

Literal  equation,  200. 

Literal  numbers,  2. 

Lowest  common  multiple,  167. 

M,  the  symbol,  48. 

Members  of  equation,  7. 

Minuend,  41. 

Monomial,    34 ;     addition    of,    35 ; 

cube  of,  111  ;  cube  root  of,  112  ; 

division  of,  88  ;  multiplication  of, 

67  ;   square  of,  110  ;   square  root 

of,  U2. 


Negative  numbers,  26  ;  addition  of, 
27  ;  division  of,  86  ;  multiplica- 
tion of,  30  ;  powers  of,  32  ;  sub- 
traction of,  43. 

Negative  term,  34  ;  exponent,  307. 

Number,  complex,  280  ;  imaginary, 
278  ;  literal,  2 ;  negative,  26 ; 
positive,  26  ;  prime,  110 ;  real, 
278  ;  unknown,  7. 

Numerical  coefficient,  34. 

Numerical  value,  17. 

Opposite  quantities,  23. 
Ordinate,  212. 
Origin,  211. 

Parallelogram,  20. 

Parentheses,  4  ;  inclosing  terms  in, 
59 ;  removing,  55. 

Perfect,  cube,  112;  square,  112; 
square  trinomial,  120. 

Periods,  247. 

Polynomial,  38 ;  arranging  a,  39. 

Polynomials,  addition  of,  38  ;  divi- 
sion of,  91 ;  factoring  of,  286;  fac- 
toring of  by  grouping,  287  ;  multi- 
plication of,  72 ;  square  root  of, 
245  ;  subtraction,  of,  45. 

Positive  number,  26  ;  quantity,  24  ; 
term,  34. 

Power,  18. 

Powers,  ascending,  39 ;  descending, 
39. 

Prime  number,  110. 

Proportion,  291  ;  by  alternation, 
294 ;  by  composition,  294 ;  by 
division,  294  ;  by  composition  and 
division,  296 ;  by  inversion, 
294. 

Proportional,  fourth,  292 ;  mean, 
292 ;  third,  292. 


INDEX 


325 


Pure  quadratic,  254. 
Pyramid,  22. 

Quadratic  equation,  148,  254 ;  com- 
plete, 258  ;  graph  of,  259  ;  hav- 
ing two  unknowns,  275  ;  imagi- 
nary roots  of,  278 ;  pure,  254 ;  solu- 
tion of  by  completing  the  square, 
263  ;  solution  of  by  factoring,  148, 
258  ;  solution  of  by  formula,  267. 

Quadratic  surd,  252. 

Quantities,  opposite,  23  ;  signed,  24. 

Quantity,  negative,  24  ;  positive,  24. 

Radical,  index  of,  113. 

Radicals,  addition  of,  252. 

Ratio,  2<)0. 

Right  angle,  15. 

Right  member,  7. 

Right  triangle,  256. 

Root,  cube,  112  ;  of  a  fraction,  251 ; 
of  an  equation,  97  ;  of  a  number, 
247;  principal,  252;  square,  112. 

Roots,  imaginary,  278  ;  of  a  quad- 
ratic, 148,  254. 

S,  the  symbol,  48. 

Signed  quantities,  24. 

Signs,  change  of,  in  an  equation,  99  ; 
law  of,  in  addition,  27  ;  in  a  frac- 
tion, 163  ;  in  division,  86  ;  in  mul- 
tiplication, 31. 

Similar  terms,  35. 


Simultaneous  equations,  218,  222. 
Solution  of  simultaneous  equations, 

221. 
Square  root,  approximate,  250  ;  by 

division,  245  ;  by  inspection,  244  ; 

of  a  monomial,  112  ;  of  a  number, 

247  ;  of  a  polynomial,  245  ;  of  a 

trinomial,  120. 
Straight  angle,  15. 
Subtrahend,  41. 
Supplement,  16. 
Supplementary  angles,  16. 
Surd,  quadratic,  252 ;   addition  of, 

252. 
Symbols  of  grouping,  55. 

Table  of  square  roots,  250. 

Term,  34  ;  degree  of,  154  ;  negative, 

34  ;  positive,  34. 
Terms,    dissimilar,    35 ;    like,    35  ; 

similar,  35  ;  unlike,  35. 
Transposition,  98. 
Triangle,  altitude  of,  21  ;   area  of, 

21 ;  base  of,  21. 

Unit,  imaginary,  279. 
Unknown  number,  7. 
Unlike  terms,  35. 

Variables,  216. 
Vertical  axis,  211. 
Vinculum,  55. 


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